Vipul: Created page with "{{square matrix property}} ==Definition== ===Verbal definition=== A '''scalar matrix''' can be defined in the following equivalent ways: * It is a diagonal matrix wher..."

2014-04-29T20:44:36Z

Created page with "{{square matrix property}} ==Definition== ===Verbal definition=== A '''scalar matrix''' can be defined in the following equivalent ways: * It is a diagonal matrix wher..."

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{{square matrix property}}

==Definition==

===Verbal definition===

A '''scalar matrix''' can be defined in the following equivalent ways:

* It is a [[diagonal matrix]] where all the diagonal entries are equal to one another.
* It is a scalar multiple of the [[identity matrix]].

===Algebraic definition===

Suppose <math>n</math> is a positive integer. A <math>n \times n</math> matrix <math>A = (a_{ij})</math> is termed a '''scalar matrix''' if '''both''' the following hold:

* Non-diagonal entries are zero: <math>a_{ij} = 0 \ \forall i \ne j \in \{ 1,2,\dots,n \}</math>
* Diagonal entries are equal to one another: <math>a_{ii} = a_{jj} \ \forall \ i,j \in \{ 1,2,\dots,n\}</math>

==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::diagonal matrix]] || all off-diagonal entries are zero, but the diagonal entries may differ from one another || || The matrix <math>\begin{pmatrix} 1 & 0 \\ 0 & 0 \\\end{pmatrix}</math> is diagonal but not scalar. || {{intermediate notions short|diagonal matrix|scalar matrix}}
|-
| [[Stronger than::Toeplitz matrix]] || each row is obtained as a cyclic shift (by one to the right) of the preceding row || || A matrix of the form <math>\begin{pmatrix} a & b \\ b & a \\\end{pmatrix}</math> || {{intermediate notions short|Toeplitz matrix|scalar matrix}}
|-
| [[Stronger than::symmetric matrix]] || equals its [[matrix tranpose]] || (via diagonal matrix) || (via diagonal matrix) || {{intermediate notions short|symmetric matrix|scalar matrix}}
|}

Scalar matrix - Revision history

Vipul: Created page with "{{square matrix property}} ==Definition== ===Verbal definition=== A '''scalar matrix''' can be defined in the following equivalent ways: * It is a diagonal matrix wher..."