Orthogonal matrix - Revision history

Vipul: /* Algebraic definition */

2015-04-17T08:37:55Z

Algebraic definition

← Older revision		Revision as of 08:37, 17 April 2015
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	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 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 ~~rows~~ have unit norm'': <math>\sum_{i=1}^n a_{ij}^2 = 1</math> for any <math>j \in \{ 1,2,\dots,n \}</math>		* ''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 ~~rows~~ 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>.		* ''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===		===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.		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.

Vipul at 06:25, 1 May 2014

2014-05-01T06:25:45Z

← Older revision		Revision as of 06:25, 1 May 2014
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	{{square matrix property}}		{{real square matrix property}}

	==Definition==		==Definition==

Vipul at 05:47, 1 May 2014

2014-05-01T05:47:55Z

← Older revision		Revision as of 05:47, 1 May 2014
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	* ''The rows have unit norm'': <math>\sum_{i=1}^n a_{ij}^2 = 1</math> for any <math>j \in \{ 1,2,\dots,n \}</math>		* ''The rows have unit norm'': <math>\sum_{i=1}^n a_{ij}^2 = 1</math> for any <math>j \in \{ 1,2,\dots,n \}</math>
	* ''Distinct rows 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>.		* ''Distinct rows 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.

Vipul: Created page with "{{square matrix property}} ==Definition== ===Verbal definition=== A square matrix is termed an '''orthogonal matrix''' if its matrix transpose equals its inverse..."

2014-05-01T05:46:36Z

Created page with "{{square matrix property}} ==Definition== ===Verbal definition=== A square matrix is termed an '''orthogonal matrix''' if its matrix transpose equals its inverse..."

New page

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