Linear - User contributions [en]

User:Vipul/Sandbox

2024-10-06T04:13:59Z

Vipul:

* <math>\sqrt{7 + 2}!! + 3 = 723</math>
* <math>2^{8 - 1} = 128</math>
* <math>2^7 - 1 = 127</math>
* <math>(7 + 2)^{\sqrt{9}} = 729</math>
* <math>9^{\sqrt{7 + 2}} = 729</math>
* <math>(3 + 4)^3 = 343</math>
* <math>6^{2 + 1} = 216</math>

MediaWiki:Sitenotice

2024-10-06T04:05:50Z

Vipul:

Want site search autocompletion? See [[Project:Enabling site search autocompletion|here]] 
Encountering 429 Too Many Requests errors when browsing the site? See [[Project:429 Too Many Requests error|here]]

User:Vipul/Sandbox

2024-10-06T04:05:26Z

Vipul:

MediaWiki:Sitenotice

2024-09-30T01:13:41Z

Vipul:

'''This site is in the process of being migrated to a new server. Edits made until this notice has been removed may be lost.''' 
Want site search autocompletion? See [[Project:Enabling site search autocompletion|here]] 
Encountering 429 Too Many Requests errors when browsing the site? See [[Project:429 Too Many Requests error|here]]

MediaWiki:Sitenotice

2024-08-26T00:00:24Z

Vipul:

File:Site search autocompletion working.png

2024-08-25T23:59:47Z

Vipul:

File:Site search autocompletion broken.png

2024-08-25T23:59:22Z

Vipul:

Linear:Enabling site search autocompletion

2024-08-25T23:56:17Z

Vipul: Created page with "Content copied from Ref:Ref:Enabling site search autocompletion. Images used are specific to this site (Linear). Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on. ==What's wrong with site search autocompletion and how to fix it== ===What's wrong=== When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in th..."

Content copied from [[Ref:Ref:Enabling site search autocompletion]]. Images used are specific to this site (Linear).

Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on.

==What's wrong with site search autocompletion and how to fix it==

===What's wrong===

When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in the screenshot below:

[[File:Site search autocompletion broken.png]]

Note that the actual search is still working -- you just have to hit Enter after typing the search query and it'll go to the search results page. It's the autocompletion before you hit Enter that is broken.

===How to fix it===

To fix it, you need to follow these steps:

* Write to vipulnaik1@gmail.com asking for a login to the site. Please include the following with your request: preferred username, preferred initial password (you can change it after logging in), real name (if you want it entered), email address to use (if you want an actual email address by which you can be contacted), and whether you want edit access as well. You don't need edit access for enabling site search autocompletion.
* Log in to the site. Then go to [[Special:Preferences]]. Go to the Appearance section and switch the Skin from "Vector (2022)" to "Vector legacy (2010)".
* Make sure to hit "Save" at the bottom.
* Now you can reload the page or load a new page.

Site search autocompletion should now work. Here's an example:

[[File:Site search autocompletion working.png]]

==More background==

We've recently upgraded the MediaWiki version of this wiki from 1.35.13 to 1.41.2 (see [[Special:Version]]). The upgrade allows us to migrate the wiki to a more modern operating system version running PHP 8. With the current setup for MediaWiki 1.41.2, we're in this situation:

* The "Vector legacy (2010)" skin has site search autocompletion working, but it doesn't render well on small screens. Specifically, even on small mobile screens, it still shows the left menu, and doesn't properly use the MobileFrontend extension settings.
* The "Vector (2022)" skin doesn't have site search autocompletion working (see screenshots in preceding section) but it does render fine on mobile devices.

It is possible to set only one default skin (that is applicable to all non-logged-in users and is the default for logged-in users who have not configured a skin for themselves). So, the selection of default skin comes down to whether it's more important for casual users to have the mobile experience working or to have site search autocompletion working. Based on a general understanding of user behavior, we believe that having a usable mobile experience is more important for casual users than having site search autocompletion.

However, for power users who are using the site extensively, site search autocompletion may be important. That's why we've written this page giving guidance on how to set up site search autocompletion.

Linear:429 Too Many Requests error

2024-08-25T23:55:28Z

Vipul: Created page with "This content is copied from Ref:Ref:429 Too Many Requests error. If you get a 429 Too Many Requests error when browsing this site, read on. You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work. If you are an actual h..."

This content is copied from [[Ref:Ref:429 Too Many Requests error]].

If you get a 429 Too Many Requests error when browsing this site, read on.

You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work.

If you are an actual human being with a legitimate reason to be browsing the site heavily, first, thank you and sorry about this! We set rate limits to prevent bots, spiders, spammers, and malicious actors from consuming too much of our server's resources so that our server's resources can be devoted to real humans like you. Consider writing to vipulnaik1@gmail.com with your IP address to have the IP address whitelisted. You can get your IP address by [https://www.google.com/search?q=my+ip+address Googling "my IP address"] (scroll down a little bit to where Google includes the IP address in a box). NOTE: If you have both an IPv4 address and an IPv6 address, you should send both; the server supports both IPv4 and IPv6, so either may end up getting used. To check if you have an IPv6 address, try visiting [https://ipv6.google.com/ ipv6.google.com].

If your IP address changes, or you are away from your home network, then you'll get rate-limited again. So if you find yourself getting rate-limited after already having been whitelisted, check if you are on a different IP address than the one for which you requested whitelisting.

MediaWiki:Sitenotice

2024-08-25T23:51:22Z

Vipul: Blanked the page

User:Vipul/Sandbox

2024-08-25T23:50:47Z

Vipul:

* <math>\sqrt{7 + 2}!! + 3 = 723</math>
* <math>2^{8 - 1} = 128</math>
* <math>2^7 - 1 = 127</math>
* <math>(7 + 2)^{\sqrt{9}} = 729</math>
* <math>9^{\sqrt{7 + 2}} = 729</math>

User:Vipul/Sandbox

2024-08-25T23:46:33Z

Vipul:

* <math>\sqrt{7 + 2}!! + 3 = 723</math>
* <math>2^{8 - 1} = 128</math>
* <math>2^7 - 1 = 127</math>
* <math>(7 + 2)^{\sqrt{9}} = 729</math>

User:Vipul/Sandbox

2024-08-25T23:41:40Z

Vipul:

* <math>\sqrt{7 + 2}!! + 3 = 723</math>
* <math>2^{8 - 1} = 128</math>
* <math>2^7 - 1 = 127</math>

MediaWiki:Sitenotice

2024-08-25T23:35:08Z

Vipul: Created page with "'''This wiki is in the process of being upgraded. The site may go down intermittently. Please try to avoid editing until this notice has been removed.'''"

'''This wiki is in the process of being upgraded. The site may go down intermittently. Please try to avoid editing until this notice has been removed.'''

User:Vipul/Sandbox

2024-07-20T05:46:27Z

Vipul:

* <math>\sqrt{7 + 2}!! + 3 = 723</math>
* <math>2^{8 - 1} = 128</math>

User:Vipul/Sandbox

2024-07-20T05:36:36Z

Vipul: Created page with "* <math>\sqrt{7 + 2}!! + 3 = 723</math>"

* <math>\sqrt{7 + 2}!! + 3 = 723</math>

Orthogonal matrix

2015-04-17T08:37:55Z

Vipul: /* Algebraic definition */

{{real square matrix property}}

==Definition==

===Verbal definition===

A [[square matrix]] is termed an '''orthogonal matrix''' if its [[matrix transpose]] equals its [[inverse matrix]]. In symbols, a <math>n \times n</math> matrix <math>A</math> is termed an orthogonal matrix if <math>A^T = A^{-1}</math>, or equivalently, <math>AA^T = I_n</math>, the <math>n \times n</math> identity matrix.

===Algebraic definition===

Suppose <math>n</math> is a positive integer. A <math>n \times n</math> square matrix <math>A = (a_{ij})_{1 \le i \le n, 1 \le j \le n}</math> is termed an '''orthogonal matrix''' if the following hold:

* ''The rows have unit norm'': <math>\sum_{j=1}^n a_{ij}^2 = 1</math> for any <math>i \in \{ 1,2,\dots,n \}</math>
* ''Distinct rows are orthogonal'': <math>\sum_{j=1}^n a_{ij}a_{kj} = 0</math> for any <math>i \ne k \in \{ 1,2,\dots,n\}</math>.

The above conditions are framed in terms of the rows, but an equivalent formulation in terms of the columns works (this is not ''a priori'' obvious, and follows from the involutive nature of the transpose and inverse operations). Explicitly:

* ''The columns have unit norm'': <math>\sum_{i=1}^n a_{ij}^2 = 1</math> for any <math>j \in \{ 1,2,\dots,n \}</math>
* ''Distinct columns are orthogonal'': <math>\sum_{i=1}^n a_{ij}a_{ik} = 0</math> for any <math>j \ne k \in \{ 1,2,\dots,n\}</math>.

===Significance of underlying ring===

The definition of orthogonal matrix presented here makes sense over any commutative unital ring. However, it has particular relevance for matrices where the entries are restricted to the reals or a subring of the reals.

Log-determinant function

2014-05-26T01:35:25Z

Vipul: Created page with "==Definition== The '''log-determinant function''', sometimes denoted <math>\operatorname{logdet}</math>, is a function from the set of symmetric square..."

==Definition==

The '''log-determinant function''', sometimes denoted <math>\operatorname{logdet}</math>, is a function from the set of [[symmetric matrix|symmetric]] [[square matrix|square matrices]] with real entries to the set <math>[0,\infty]</math> (nonnegative real numbers along with infinity) defined as follows:

* If the matrix is a [[symmetric positive-definite matrix]], the log-determinant is defined as the logarithm of the determinant of the matrix. Equivalently, it is the sum of the logarithms of the eigenvalues of the matrix, all of which are positive real numbers (note that repeated eigenvalues are counted with multiplicity).
* If the matrix is not positive-definite, the log-determinant is defined as <math>+\infty</math>.

Logdet

2014-05-26T01:32:56Z

Vipul: Redirected page to Log-determinant function

#redirect [[Log-determinant function]]

Frobenius norm

2014-05-15T03:18:21Z

Vipul:

==Definition==

===For a matrix with real entries===

Suppose <math>m,n</math> are positive integers and <math>A</math> is a <math>m \times n</math> matrix. The '''Frobenius norm''' of <math>A</math>, denoted <math>\| A \|_F</math>, can be defined in the following equivalent ways:

# It is the square root of the sum of squares of all the entries of <math>A</math>, i.e., it is the sum <math>\sqrt{\sum_{i=1}^m \sum_{j=1}^n a_{ij}^2}</math>.
# It is the square root of the trace of the <math>m \times m</math> matrix <math>AA^T</math>, where <math>A^T</math> is the [[matrix transpose]] of <math>A</math>.
# It is the square root of the trace of the <math>n \times n</math> matrix <math>A^TA</math>, where <math>A^T</math> is the [[matrix transpose]] of <math>A</math>.
# It is the square root of the sum of squares of the <math>\min \{m, n \}</math> many [[singular value]]s of <math>A</math>.

The Frobenius norm is invariant under orthogonal transformations (and in particular, under rotations) and is an easy-to-compute invariant.

===For a matrix with complex entries===

Suppose <math>m,n</math> are positive integers and <math>A</math> is a <math>m \times n</math> matrix. The '''Frobenius norm''' of <math>A</math>, denoted <math>\| A \|_F</math>, can be defined in the following equivalent ways:

# It is the square root of the sum of squares of the moduli of the entries of <math>A</math>, i.e., it is the sum <math>\sum_{i=1}^m \sum_{j=1}^n |a_{ij}|^2</math>.
# It is the square root of the trace of the matrix <math>AA^*</math> where <math>A^*</math> is the [[matrix conjugate transpose]] of <math>A</math>.
# It is the square root of the trace of the matrix <math>A^*A</math> where <math>A^*</math> is the [[matrix conjugate transpose]] of <math>A</math>.
# It is the square root of the sum of squares of the <math>\min \{m, n \}</math> many [[singular value]]s of <math>A</math>.

Frobenius norm

2014-05-13T02:45:46Z

Vipul: /* Definition */

==Definition==

===For a matrix with real entries===

Suppose <math>m,n</math> are positive integers and <math>A</math> is a <math>m \times n</math> matrix. The '''Frobenius norm''' of <math>A</math>, denoted <math>\| A \|_F</math>, can be defined in the following equivalent ways:

# It is the square root of the sum of squares of all the entries of <math>A</math>, i.e., it is the sum <math>\sqrt{\sum_{i=1}^m \sum_{j=1}^n a_{ij}^2}</math>.
# It is the square root of the trace of the <math>m \times m</math> matrix <math>AA^T</math>, where <math>A^T</math> is the [[matrix transpose]] of <math>A</math>.
# It is the square root of the trace of the <math>n \times n</math> matrix <math>A^TA</math>, where <math>A^T</math> is the [[matrix transpose]] of <math>A</math>.
# It is the [[root mean square]] of the <math>\min \{m, n \}</math> many [[singular value]]s of <math>A</math>.

The Frobenius norm is invariant under orthogonal transformations (and in particular, under rotations) and is an easy-to-compute invariant.

===For a matrix with complex entries===

Suppose <math>m,n</math> are positive integers and <math>A</math> is a <math>m \times n</math> matrix. The '''Frobenius norm''' of <math>A</math>, denoted <math>\| A \|_F</math>, can be defined in the following equivalent ways:

# It is the square root of the sum of squares of the moduli of the entries of <math>A</math>, i.e., it is the sum <math>\sum_{i=1}^m \sum_{j=1}^n |a_{ij}|^2</math>.
# It is the square root of the trace of the matrix <math>AA^*</math> where <math>A^*</math> is the [[matrix conjugate transpose]] of <math>A</math>.
# It is the square root of the trace of the matrix <math>A^*A</math> where <math>A^*</math> is the [[matrix conjugate transpose]] of <math>A</math>.
# It is the [[root mean square]] of the <math>\min \{m, n \}</math> many [[singular value]]s of <math>A</math>.

NLLS

2014-05-09T16:51:54Z

Vipul: Redirected page to Non-linear least squares

#redirect [[Non-linear least squares]]

Frobenius norm

2014-05-09T16:36:45Z

Vipul:

==Definition==

===For a matrix with real entries===

Suppose <math>m,n</math> are positive integers and <math>A</math> is a <math>m \times n</math> matrix. The '''Frobenius norm''' of <math>A</math>, denoted <math>\| A \|_F</math>, can be defined in the following equivalent ways:

# It is the sum of squares of all the entries of <math>A</math>, i.e., it is the sum <math>\sum_{i=1}^m \sum_{j=1}^n a_{ij}^2</math>.
# It is the trace of the <math>m \times m</math> matrix <math>AA^T</math>, where <math>A^T</math> is the [[matrix transpose]] of <math>A</math>.
# It is the trace of the <math>n \times n</math> matrix <math>A^TA</math>, where <math>A^T</math> is the [[matrix transpose]] of <math>A</math>.
# It is the [[root mean square]] of the <math>\min \{m, n \}</math> many [[singular value]]s of <math>A</math>.

The Frobenius norm is invariant under orthogonal transformations (and in particular, under rotations) and is an easy-to-compute invariant.

===For a matrix with complex entries===

Suppose <math>m,n</math> are positive integers and <math>A</math> is a <math>m \times n</math> matrix. The '''Frobenius norm''' of <math>A</math>, denoted <math>\| A \|_F</math>, can be defined in the following equivalent ways:

# It is the sum of squares of the moduli of the entries of <math>A</math>, i.e., it is the sum <math>\sum_{i=1}^m \sum_{j=1}^n |a_{ij}|^2</math>.
# It is the trace of the matrix <math>AA^*</math> where <math>A^*</math> is the [[matrix conjugate transpose]] of <math>A</math>.
# It is the trace of the matrix <math>A^*A</math> where <math>A^*</math> is the [[matrix conjugate transpose]] of <math>A</math>.
# It is the [[root mean square]] of the <math>\min \{m, n \}</math> many [[singular value]]s of <math>A</math>.

Frobenius norm

2014-05-09T16:35:54Z

Vipul:

==Definition==

===For a matrix with real entries===

Suppose <math>m,n</math> are positive integers and <math>A</math> is a <math>m \times n</math> matrix. The '''Frobenius norm''' of <math>A</math>, denoted <math>\| A \|_F</math>, can be defined in the following equivalent ways:

# It is the sum of squares of all the entries of <math>A</math>, i.e., it is the sum <math>\sum_{i=1}^m \sum_{j=1}^n a_{ij}^2</math>.
# It is the trace of the <math>m \times m</math> matrix <math>AA^T</math>, where <math>A^T</math> is the [[matrix transpose]] of <math>A</math>.
# It is the trace of the <math>n \times n</math> matrix <math>A^TA</math>, where <math>A^T</math> is the [[matrix transpose]] of <math>A</math>.

The Frobenius norm is invariant under orthogonal transformations (and in particular, under rotations) and is an easy-to-compute invariant.

===For a matrix with complex entries===

Suppose <math>m,n</math> are positive integers and <math>A</math> is a <math>m \times n</math> matrix. The '''Frobenius norm''' of <math>A</math>, denoted <math>\| A \|_F</math>, can be defined in the following equivalent ways:

# It is the sum of squares of the moduli of the entries of <math>A</math>, i.e., it is the sum <math>\sum_{i=1}^m \sum_{j=1}^n |a_{ij}|^2</math>.
# It is the trace of the matrix <math>AA^*</math> where <math>A^*</math> is the [[matrix conjugate transpose]] of <math>A</math>.
# It is the trace of the matrix <math>A^*A</math> where <math>A^*</math> is the [[matrix conjugate transpose]] of <math>A</math>.

Frobenius norm

2014-05-09T16:16:14Z

Vipul: Created page with "==Definition== ===For a matrix with real entries=== Suppose <math>m,n</math> are positive integers and <math>A</math> is a <math>m \times n</math> matrix. The '''Frobenius n..."

==Definition==

===For a matrix with real entries===

Suppose <math>m,n</math> are positive integers and <math>A</math> is a <math>m \times n</math> matrix. The '''Frobenius norm''' of <math>A</math>, denoted <math>\| A \|_F</math>, can be defined in the following equivalent ways:

# It is the sum of squares of all the entries of <math>A</math>, i.e., it is the sum <math>\sum_{i=1}^m \sum_{j=1}^n a_{ij}^2</math>.
# It is the trace of the <math>m \times m</math> matrix <math>AA^T</math>, where <math>A^T</math> is the [[matrix transpose]] of <math>A</math>.
# It is the trace of the <math>n \times n</math> matrix <math>A^TA</math>, where <math>A^T</math> is the [[matrix transpose]] of <math>A</math>.

The Frobenius norm is invariant under orthogonal transformations (and in particular, under rotations) and is an easy-to-compute invariant.

===For a matrix with complex entries===

Suppose <math>m,n</math> are positive integers and <math>A</math> is a <math>m \times n</math> matrix. The '''Frobenius norm''' of <math>A</math>, denoted <math>\| A \|_F</math>, can be defined in the following equivalent ways:

# It is the sum of squares of the moduli of the entries of <math>A</math>, i.e., it is the sum <math>\sum_{i=1}^m \sum_{j=1}^n |a_{ij}|^2</math>.
# It is the trace of the matrix <math>AA^*</math>.
# It is the trace of the matrix <math>A^*A</math>.

Sherman-Morrison formula

2014-05-09T16:06:18Z

Vipul: Created page with "==Definition== Suppose <math>n</math> is a positive integer, <math>A</math> is an invertible <math>n \times n</math> matrix, and <math>\vec{u},\vec{v}</math> are <math>n</mat..."

==Definition==

Suppose <math>n</math> is a positive integer, <math>A</math> is an invertible <math>n \times n</math> matrix, and <math>\vec{u},\vec{v}</math> are <math>n</math>-dimensional column vectors. The [[Hadamard product]] <math>\vec{u}\vec{v}^T</math> is therefore a <math>n \times n</math> matrix of rank one. We have the following:

* The matrix <math>A + \vec{u}\vec{v}^T</math> is invertible if and only if the real number <math>1 + \vec{v}^TA^{-1}\vec{u}</math> is nonzero.
* If the condition above holds, we have the formula:

<math>(A + \vec{u}\vec{v}^T)^{-1} = A^{-1} - \frac{A^{-1}\vec{u}\vec{v}^TA^{-1}}{1 + \vec{v}^TA^{-1}\vec{u}}</math>

Reduced row echelon form

2014-05-01T18:20:01Z

Vipul: /* The reduced row echelon form of a matrix */

{{matrix property}}

==Definition==

===A matrix being in reduced row echelon form===

A matrix is said to be a '''reduced row echelon matrix''', or said to be in '''reduced row echelon form''' ('''rref'''), if it satisfies the following conditions:

* All nonzero rows are above all zero rows. Here, a ''nonzero row'' is a row that has at least one nonzero entry, and a zero row is a row where ''all'' entries are zero.
* The first nonzero entry in any nonzero row occurs in a strictly later column than the first nonzero entry in the row immediately above it (and hence also, in all the rows above it).
* The first nonzero entry in any nonzero row is 1 (this condition is omitted in some definitions) ''and all other entries in the column of that entry are zero''.

With the exception of the (emphasized) second half of the last condition, the conditions above define [[row echelon form]].

===The reduced row echelon form of a matrix===

For a <math>m \times n</math> matrix <math>A</math>, the reduced row echelon form of <math>A</math> is the unique matrix <math>B</math> in reduced row echelon form such that we can write <math>B = SA</math> where <math>S</math> is a <math>m \times m</math> [[invertible matrix]].

Reduced row echelon form

2014-05-01T18:19:13Z

Vipul:

{{matrix property}}

==Definition==

===A matrix being in reduced row echelon form===

A matrix is said to be a '''reduced row echelon matrix''', or said to be in '''reduced row echelon form''' ('''rref'''), if it satisfies the following conditions:

* All nonzero rows are above all zero rows. Here, a ''nonzero row'' is a row that has at least one nonzero entry, and a zero row is a row where ''all'' entries are zero.
* The first nonzero entry in any nonzero row occurs in a strictly later column than the first nonzero entry in the row immediately above it (and hence also, in all the rows above it).
* The first nonzero entry in any nonzero row is 1 (this condition is omitted in some definitions) ''and all other entries in the column of that entry are zero''.

With the exception of the (emphasized) second half of the last condition, the conditions above define [[row echelon form]].

===The reduced row echelon form of a matrix===

For a <math>m \times n</math> matrix </math>A</math>, the reduced row echelon form of <math>A</math> is the unique matrix <math>B</math> in reduced row echelon form such that we can write <math>B = SA</math> where <math>S</math> is a <math>m \times m</math> [[invertible matrix]].

Reduced row echelon form

2014-05-01T18:18:57Z

Vipul: Created page with "{{matrix property}} ===Definition== ===A matrix being in reduced row echelon form=== A matrix is said to be a '''reduced row echelon matrix''', or said to be in '''reduced..."

{{matrix property}}

===Definition==

===A matrix being in reduced row echelon form===

A matrix is said to be a '''reduced row echelon matrix''', or said to be in '''reduced row echelon form''' ('''rref'''), if it satisfies the following conditions:

* All nonzero rows are above all zero rows. Here, a ''nonzero row'' is a row that has at least one nonzero entry, and a zero row is a row where ''all'' entries are zero.
* The first nonzero entry in any nonzero row occurs in a strictly later column than the first nonzero entry in the row immediately above it (and hence also, in all the rows above it).
* The first nonzero entry in any nonzero row is 1 (this condition is omitted in some definitions) ''and all other entries in the column of that entry are zero''.

With the exception of the (emphasized) second half of the last condition, the conditions above define [[row echelon form]].

===The reduced row echelon form of a matrix===

For a <math>m \times n</math> matrix </math>A</math>, the reduced row echelon form of <math>A</math> is the unique matrix <math>B</math> in reduced row echelon form such that we can write <math>B = SA</math> where <math>S</math> is a <math>m \times m</math> [[invertible matrix]].

Rref

2014-05-01T18:15:33Z

Vipul: Redirected page to Reduced row echelon form

#redirect [[reduced row echelon form]]

Ref

2014-05-01T18:14:20Z

Vipul: Redirected page to Row echelon form

#redirect [[row echelon form]]

Row echelon matrix

2014-05-01T18:14:03Z

Vipul: Redirected page to Row echelon form

#redirect [[row echelon form]]

Row echelon form

2014-05-01T18:13:42Z

Vipul: Created page with "{{matrix property}} ==Definition== A matrix is said to be a '''row echelon matrix''', or is said to be in '''row echelon form''', if it satisfies the following conditions:..."

{{matrix property}}

==Definition==

A matrix is said to be a '''row echelon matrix''', or is said to be in '''row echelon form''', if it satisfies the following conditions:

* All nonzero rows are above all zero rows. Here, a ''nonzero row'' is a row that has at least one nonzero entry, and a zero row is a row where ''all'' entries are zero.
* The first nonzero entry in any nonzero row occurs in a strictly later column than the first nonzero entry in the row immediately above it (and hence also, in all the rows above it).
* The first nonzero entry in any nonzero row is 1 (this condition is omitted in some definitions).

==Relation with other properties==

===Stronger properties===

* [[Weaker than::reduced row echelon form]]

Permutation matrix

2014-05-01T18:06:53Z

Vipul: /* Weaker properties */

{{square matrix property}}

==Definition==

A [[square matrix]] is termed a '''permutation matrix''' if it satisfes '''both''' these conditions:

# Every row has exactly one nonzero entry, and the value of that entry is 1.
# Every column has exactly one nonzero entry, and the value of that entry is 1.

Note that assuming that the matrix is an [[invertible matrix]], the two conditions are equivalent (and moreover, all permutation matrices are invertible). Thus, a permutation matrix can be defined as an invertible matrix satisfying condition (1), or equivalently, as an invertible matrix satisfying condition (2).

==Relation with other properties==

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::monomial matrix]] || || || || {{intermediate notions short|monomial matrix|permutation matrix}}
|-
| [[Stronger than::signed permutation matrix]] || || || || {{intermediate notions short|signed permutation matrix|permutation matrix}}
|-
| [[Stronger than::doubly stochastic matrix]] || || || || {{intermediate notions short|doubly stochastic matrix|permutation matrix}}
|-
| [[Stronger than::row-stochastic matrix]] || || || || {{intermediate notions short|row-stochastic matrix|permutation matrix}}
|-
| [[Stronger than::column-stochastic matrix]] || || || || {{intermediate notions short|column-stochastic matrix|permutation matrix}}
|}

Permutation matrix

2014-05-01T18:06:38Z

Vipul: Created page with "{{square matrix property}} ==Definition== A square matrix is termed a '''permutation matrix''' if it satisfes '''both''' these conditions: # Every row has exactly one n..."

Upper triangular matrix

2014-05-01T17:54:57Z

Vipul: /* Stronger properties */

{{square matrix property}}

==Definition==

===Verbal definition===

A [[square matrix]] is termed an '''upper triangular matrix''' if all its entries ''below'' the [[main diagonal]] are zero.

===Algebraic description===

Suppose <math>n</math> is a positive integer. A <math>n \times n</math> [[square matrix]] <math>A = (a_{ij})_{1 \le i \le n, 1 \le j \le n}</math> is termed an '''upper triangular matrix''' if the following holds:

<math>a_{ij} = \ \forall \ i > j \in \{ 1,2,\dots,n\}</math>

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::scalar matrix]] || scalar multiple of [[identity matrix]] || (via diagonal matrix) || (via diagonal matrix) || {{intermediate notions short|upper triangular matrix|scalar matrix}}
|-
| [[Weaker than::diagonal matrix]] || all entries not on the main diagonal are zero || || the matrix <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math> is a counterexample || {{intermediate notions short|upper triangular matrix|diagonal matrix}}
|-
| [[Weaker than::upper unitriangular matrix]] || upper triangular matrix with all diagonal entries equal to 1 || || || {{intermediate notions short|upper triangular matrix|upper unitriangular matrix}}
|-
| [[Weaker than::strictly upper triangular matrix]] || all entries on ''and'' below the main diagonal are zero || || || {{intermediate notions short|upper triangular matrix|strictly upper triangular matrix}}
|}

===Contrast properties===

* [[Lower triangular matrix]]: A matrix is upper triangular and lower triangular if and only if it is a [[diagonal matrix]].
* [[Orthogonal matrix]]
* [[Symmetric matrix]]
* [[Skew-symmetric matrix]]

Upper triangular matrix

2014-05-01T17:54:41Z

Vipul: /* Contrast properties= */

{{square matrix property}}

==Definition==

===Verbal definition===

A [[square matrix]] is termed an '''upper triangular matrix''' if all its entries ''below'' the [[main diagonal]] are zero.

===Algebraic description===

Suppose <math>n</math> is a positive integer. A <math>n \times n</math> [[square matrix]] <math>A = (a_{ij})_{1 \le i \le n, 1 \le j \le n}</math> is termed an '''upper triangular matrix''' if the following holds:

<math>a_{ij} = \ \forall \ i > j \in \{ 1,2,\dots,n\}</math>

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::scalar matrix]] || scalar multiple of [[identity matrix]] || (via diagonal matrix) || (via diagonal matrix) || {{intermediate notions short|upper triangular matrix|scalar matrix}}
|-
| [[Weaker than::diagonal matrix]] || all entries not on the main diagonal are zero || || the matrix <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\\end{pmatrix}</math> is a counterexample || {{intermediate notions short|upper triangular matrix|diagonal matrix}}
|-
| [[Weaker than::upper unitriangular matrix]] || upper triangular matrix with all diagonal entries equal to 1 || || || {{intermediate notions short|upper triangular matrix|upper unitriangular matrix}}
|-
| [[Weaker than::strictly upper triangular matrix]] || all entries on ''and'' below the main diagonal are zero || || || {{intermediate notions short|upper triangular matrix|strictly upper triangular matrix}}
|}
===Contrast properties===

* [[Lower triangular matrix]]: A matrix is upper triangular and lower triangular if and only if it is a [[diagonal matrix]].
* [[Orthogonal matrix]]
* [[Symmetric matrix]]
* [[Skew-symmetric matrix]]

Upper triangular matrix

2014-05-01T17:54:13Z

Vipul: Created page with "{{square matrix property}} ==Definition== ===Verbal definition=== A square matrix is termed an '''upper triangular matrix''' if all its entries ''below'' the main dia..."

{{square matrix property}}

==Definition==

===Verbal definition===

A [[square matrix]] is termed an '''upper triangular matrix''' if all its entries ''below'' the [[main diagonal]] are zero.

===Algebraic description===

Suppose <math>n</math> is a positive integer. A <math>n \times n</math> [[square matrix]] <math>A = (a_{ij})_{1 \le i \le n, 1 \le j \le n}</math> is termed an '''upper triangular matrix''' if the following holds:

<math>a_{ij} = \ \forall \ i > j \in \{ 1,2,\dots,n\}</math>

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::scalar matrix]] || scalar multiple of [[identity matrix]] || (via diagonal matrix) || (via diagonal matrix) || {{intermediate notions short|upper triangular matrix|scalar matrix}}
|-
| [[Weaker than::diagonal matrix]] || all entries not on the main diagonal are zero || || the matrix <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\\end{pmatrix}</math> is a counterexample || {{intermediate notions short|upper triangular matrix|diagonal matrix}}
|-
| [[Weaker than::upper unitriangular matrix]] || upper triangular matrix with all diagonal entries equal to 1 || || || {{intermediate notions short|upper triangular matrix|upper unitriangular matrix}}
|-
| [[Weaker than::strictly upper triangular matrix]] || all entries on ''and'' below the main diagonal are zero || || || {{intermediate notions short|upper triangular matrix|strictly upper triangular matrix}}
|}
==Contrast properties===

* [[Lower triangular matrix]]: A matrix is upper triangular and lower triangular if and only if it is a [[diagonal matrix]].
* [[Orthogonal matrix]]

Bandwidth of a matrix

2014-05-01T17:40:19Z

Vipul:

==Definition==

Suppose <math>n</math> is a positive integer and <math>A = (a_{ij})_{1 \le i \le n, 1 \le j \le n}</math> is a <math>n \times n</math> [[square matrix]].

The '''left half-bandwidth''' of <math>A</math> is defined as the smallest positive integer <math>k_1</math> such that <math>a_{ij} = 0</math> whenever <math>j . In other words, entries that are more than <math>k_1</math> positions below the [[main diagonal]] are zero.

The '''right half-bandwidth''' of <math>A</math> is defined as the smallest positive integer <math>k_2</math> such that <math>a_{ij} = 0</math> whenever <math>j . In other words, entries that are more than <math>k_2</math> positions above the [[main diagonal]] are zero.

The '''banwidth''' is defined as <math>k_1 + k_2 + 1</math>, where <math>k_1,k_2</math> are the left and right half-bandwidths respectively.

===Ambiguity with terminology===

When we say that a matrix has a given bandwidth (respectively, a given left half-bandwidth or a given right half-bandwidth) what we mean is that the bandwidth (respectively, the left or right half-bandwidth) is ''at most'' that quantity, not that it is necessarily exactly equal to that quantity.

===The notion of band matrix or banded matrix===

The term '''band matrix''' or '''banded matrix''' is used for a matrix whose bandwidth is reasonably small.

==Particular cases==

{| class="sortable" border="1"
! Matrix type (all matrices are <math>n \times n</math>) !! Left half-bandwidth (upper bound) !! Right half-bandwidth (upper bound) !! Bandwidth (upper bound)
|-
| [[diagonal matrix]] || 0 || 0 || 1
|-
| [[tridiagonal matrix]] || 1 || 1 || 3
|-
| [[pentadiagonal matrix]] || 2 || 2 || 5
|-
| [[upper triangular matrix]] || 0 || <math>n - 1</math> || <math>n</math>
|-
| [[lower triangular matrix]] || <math>n - 1</math> || 0 || <math>n</math>
|}

Right half-bandwidth

2014-05-01T17:18:26Z

Vipul: Redirected page to Bandwidth of a matrix

#redirect [[Bandwidth of a matrix]]

Left half-bandwidth

2014-05-01T17:17:42Z

Vipul: Redirected page to Bandwidth of a matrix

#redirect [[Bandwidth of a matrix]]

Bandwidth

2014-05-01T17:17:20Z

Vipul: Redirected page to Bandwidth of a matrix

#redirect [[Bandwidth of a matrix]]

Bandwidth of a matrix

2014-05-01T17:16:39Z

Vipul:

==Definition==

Suppose <math>n</math> is a positive integer and <math>A = (a_{ij})_{1 \le i \le n, 1 \le j \le n}</math> is a <math>n \times n</math> [[square matrix]].

The '''left half-bandwidth''' of <math>A</math> is defined as the smallest positive integer <math>k_1</math> such that <math>a_{ij} = 0</math> whenever <math>j . In other words, entries that are more than <math>k_1</math> positions below the [[main diagonal]] are zero.

The '''right half-bandwidth''' of <math>A</math> is defined as the smallest positive integer <math>k_2</math> such that <math>a_{ij} = 0</math> whenever <math>j . In other words, entries that are more than <math>k_2</math> positions above the [[main diagonal]] are zero.

The '''banwidth''' is defined as <math>k_1 + k_2 + 1</math>, where <math>k_1,k_2</math> are the left and right half-bandwidths respectively.

===Ambiguity with terminology===

When we say that a matrix has a given bandwidth (respectively, a given left half-bandwidth or a given right half-bandwidth) what we mean is that the bandwidth (respectively, the left or right half-bandwidth) is ''at most'' that quantity, not that it is necessarily exactly equal to that quantity.

==Particular cases==

{| class="sortable" border="1"
! Matrix type (all matrices are <math>n \times n</math>) !! Left half-bandwidth (upper bound) !! Right half-bandwidth (upper bound) !! Bandwidth (upper bound)
|-
| [[diagonal matrix]] || 0 || 0 || 1
|-
| [[tridiagonal matrix]] || 1 || 1 || 3
|-
| [[pentadiagonal matrix]] || 2 || 2 || 5
|-
| [[upper triangular matrix]] || 0 || <math>n - 1</math> || <math>n</math>
|-
| [[lower triangular matrix]] || <math>n - 1</math> || 0 || <math>n</math>
|}

Bandwidth of a matrix

2014-05-01T17:15:54Z

Vipul:

==Definition==

Suppose <math>n</math> is a positive integer and <math>A = (a_{ij})_{1 \le i \le n, 1 \le j \le n}</math> is a <math>n \times n</math> [[square matrix]].

The '''left half-bandwidth''' of <math>A<math> is defined as the smallest positive integer <math>k_1</math> such that <math>a_{ij} = 0</math> whenever <math>j . In other words, entries that are more than <math>k_1</math> positions below the [[main diagonal]] are zero.

The '''right half-bandwidth''' of <math>A</math> is defined as the smallest positive integer <math>k_2</math> such that <math>a_{ij} = 0</math> whenever <math>j . In other words, entries that are more than <math>k_2</math> positions above the [[main diagonal]] are zero.

The '''banwidth''' is defined as <math>k_1 + k_2 + 1</math>, where <math>k_1,k_2</math> are the left and right half-bandwidths respectively.

===Ambiguity with terminology===

When we say that a matrix has a given bandwidth (respectively, a given left half-bandwidth or a given right half-bandwidth) what we mean is that the bandwidth (respectively, the left or right half-bandwidth) is ''at most'' that quantity, not that it is necessarily exactly equal to that quantity.

==Particular cases==

{| class="sortable" border="1"
! Matrix type (all matrices are <math>n \times n</math>) !! Left half-bandwidth (upper bound) !! Right half-bandwidth (upper bound) !! Bandwidth (upper bound)
|-
| [[diagonal matrix]] || 0 || 0 || 1
|-
| [[tridiagonal matrix]] || 1 || 1 || 3
|-
| [[pentadiagonal matrix]] || 2 || 2 || 5
|-
| [[upper triangular matrix]] || 0 || <math>n - 1</math> || <math>n</math>
|-
| [[lower triangular matrix]] || <math>n - 1</math> || 0 || <math>n</math>
|}

Bandwidth of a matrix

2014-05-01T17:12:02Z

Vipul: Created page with "==Definition== Suppose <math>n</math> is a positive integer and <math>A = (a_{ij})_{1 \le i \le n, 1 \le j \le n}</math> is a <math>n \times n</math> square matrix. The..."

P-matrix

2014-05-01T16:52:51Z

Vipul: Created page with "{{real square matrix property}} ==Definition== Suppose <math>n</math> is a positive integer and <math>A</math> is a <math>n \times n</math> square matrix. We say that <math>..."

{{real square matrix property}}

==Definition==

Suppose <math>n</math> is a positive integer and <math>A</math> is a <math>n \times n</math> square matrix. We say that <math>A</math> is a '''P-matrix''' if all the [[principal minor]]s of <math>A</math> are positive.

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::symmetric positive-definite matrix]] || || || || {{intermediate notions short|P-matrix|symmetric positive-definite matrix}}
|-
| [[Weaker than::positive-definite matrix]] || || || || {{intermediate notions short|P-matrix|positive-definite matrix}}
|-
| [[Weaker than::nonsingular M-matrix]] || || || || {{intermediate notions short|P-matrix|nonsingular M-matrix}}
|}

Symmetric positive-definite matrix

2014-05-01T16:47:34Z

Vipul: Created page with "{{real square matrix property}} ==Definition== Suppose <math>n</math> is a positive integer and <math>A</math> is a <math>n \times n</math> square matrix. We say that <m..."

{{real square matrix property}}

==Definition==

Suppose <math>n</math> is a positive integer and <math>A</math> is a <math>n \times n</math> [[square matrix]]. We say that <math>A</math> is a '''symmetric positive-definite matrix''' if the following equivalent conditions hold:

# ''''Symmetric and positive-definite''': <math>A = A^T</math> (i.e., <math>A</math> is a [[defining ingredient::symmetric matrix]]: it equals its [[matrix transpose]]) and <math>A</math> is a [[defining ingredient::positive-definite matrix]]: for every <math>1 \times n</math> column vector <math>x</math>, we have that <math>x^TAx \ge 0</math>, and equality holds if and only if <math>x</math> is the zero vector (in other words, <math>x^TAx > 0</math> for all nonzero <math>1 \times n</math> column vectors <math>x</math>).
# The bilinear form on <math>\R^n</math> defined by <math>(u,v) \mapsto u^TAv</math> (where the input vectors are written as column vectors) is a [[symmetric positive-definite bilinear form]].
# There is a <math>n \times n</math> [[invertible matrix]] <math>B</math> such that <math>A = BB^T</math>.
# <math>A</math> is a symmetric matrix and a [[P-matrix]].

==Relation with other properties==

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::invertible matrix]] || || || || {{intermediate notions short|invertible matrix|symmetric positive-definite matrix}}
|-
| [[Stronger than::positive-definite matrix]] || || || || {{intermediate notions short|positive-definite matrix|symmetric positive-definite matrix}}
|-
| [[Stronger than::P-matrix]] || || || || {{intermediate notions short|P-matrix|symmetric positive-definite matrix}}
|}

Essentially nonnegative matrix

2014-05-01T16:36:20Z

Vipul: Redirected page to Metzler matrix

#redirect [[Metzler matrix]]

Quasipositive matrix

2014-05-01T16:28:54Z

Vipul: Redirected page to Metzler matrix

#redirect [[Metzler matrix]]

Metzler matrix

2014-05-01T16:28:01Z

Vipul: Created page with "{{real square matrix property}} ==Definition== ===Verbal definition=== A square matrix is termed a '''Metzler matrix'''', '''quasipositive matrix''', '''quasi-positive..."

{{real square matrix property}}

==Definition==

===Verbal definition===

A [[square matrix]] is termed a '''Metzler matrix'''', '''quasipositive matrix''', '''quasi-positive matrix''', or '''essentially nonnegative matrix''' if all its off-diagonal entries are nonnegative.

===Algebraic description===

Suppose <math>n</math> is a positive integer. A <math>n \times n</math> [[square matrix]] <math>A = (a_{ij})_{1 \le i \le n, 1 \le j \le n}</math> with real entries is termed a '''Metzler matrix'''', '''quasipositive matrix''', '''quasi-positive matrix''', or '''essentially nonnegative matrix''' if it satisfies the following condition:

<math>a_{ij} \ge 0 \ \forall i \ne j \in \{1, 2, \dots, n\}</math>

===Other equivalent descriptions===

A square matrix with real entries is a Metzler matrix if and only if it is the negative of a [[Z-matrix]]. This can be used as an alternative definition.