Symmetric matrix - Revision history

Vipul: /* Algebraic definition */

2014-05-01T05:39:12Z

Algebraic definition

← Older revision		Revision as of 05:39, 1 May 2014
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	<math>a_{ij} = a_{ji} \ \forall \ i,j \in \{ 1,2,\dots,n \}</math>		<math>a_{ij} = a_{ji} \ \forall \ i,j \in \{ 1,2,\dots,n \}</math>

	Finally, because the equality condition is symmetric in <math>i,j</math>, it suffices to check it for <math>i < j</math>, so the ~~baove~~ definition if equivalent to:		Finally, because the equality condition is symmetric in <math>i,j</math>, it suffices to check it for <math>i < j</math>, so the above definition if equivalent to:

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

Vipul: /* Algebraic definition */

2014-05-01T05:33:01Z

Algebraic definition

← Older revision		Revision as of 05:33, 1 May 2014
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	<math>a_{ij} = a_{ji} \ \forall \ i,j \in \{ 1,2,\dots,n \}</math>		<math>a_{ij} = a_{ji} \ \forall \ i,j \in \{ 1,2,\dots,n \}</math>

			Finally, because the equality condition is symmetric in <math>i,j</math>, it suffices to check it for <math>i < j</math>, so the baove definition if equivalent to:

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

	==Encoding==		==Encoding==

Vipul: /* Encoding */

2014-05-01T05:30:44Z

Encoding

← Older revision		Revision as of 05:30, 1 May 2014
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	We can encode a symmetric matrix by simply storing the entries on or above the diagonal. This is a total of <math>n(n+1)/2</math> entries instead of the total of <math>n^2</math> entries.		We can encode a symmetric matrix by simply storing the entries on or above the diagonal. This is a total of <math>n(n+1)/2</math> entries instead of the total of <math>n^2</math> entries.

			==Small cases==

			For the equal entries <math>a_{ij}</math> and <math>a_{ji}</math>, <math>i < j</math>, we will use <math>a_{ij}</math> as the label. This is purely a matter of convention.

			{\| class="sortable" border="1"
			! <math>n</math> !! General description of <math>n \times n</math> symmetric matrix !! Number of free parameters (equals <math>n(n+1)/2</math>)
			\|-
			\| 1 \|\| <math>\begin{pmatrix} a_{11} \\\end{pmatrix}</math> \|\| 1
			\|-
			\| 2 \|\| <math>\begin{pmatrix} a_{11} & a_{12} \\ a_{12} & a_{22} \\\end{pmatrix}</math> \|\| 3
			\|-
			\| 3 \|\| <math>\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{12} & a_{22} & a_{23} \\ a_{13} & a_{23} & a_{33} \\\end{pmatrix}</math> \|\| 6
			\|}

			==Matrix operations==

			===Invariant computation===

			For both the trace and the determinant, there does not seem to be a simpler way of computing them than the standard approach for square matrices.

			===Operations===

			{\| class="sortable" border="1"
			! Matrix operation !! Description in terms of encoding !! Number of additions !! Number of multiplications !! Number of inverse computations !! Set of symmetric matrices closed under operation?
			\|-
			\| Addition of two symmetric matrices \|\| Add the corresponding entries of the two matrices \|\| <math>n(n+1)/2</math> \|\| 0 \|\| 0 \|\| Yes
			\|-
			\| Multiplication of two symmetric matrices \|\| Usual matrix multiplication algorithms; symmetry does not seem to offer speedup \|\| <math>n^2(n - 1)</math> for [[naive matrix multiplication]] \|\| <math>n^3</math> for naive matrix multiplication \|\| 0 \|\| No
			\|}

	==Relation with other properties==		==Relation with other properties==

Vipul: /* Definition */

2014-05-01T05:17:34Z

Definition

← Older revision		Revision as of 05:17, 1 May 2014
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	<math>a_{ij} = a_{ji} \ \forall \ i \ne j \in \{ 1,2,\dots,n \}</math>		<math>a_{ij} = a_{ji} \ \forall \ i \ne j \in \{ 1,2,\dots,n \}</math>

			Note that the <math>i \ne j</math> condition is not required; the above definition is equivalent to:

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

	==Encoding==		==Encoding==

Vipul: Created page with "{{square matrix property}} ==Definition== ===Verbal definition=== A '''symmetric matrix''' is a square matrix that equals its own matrix transpose. Explicitly, a sq..."

2014-04-29T21:11:36Z

Created page with "{{square matrix property}} ==Definition== ===Verbal definition=== A '''symmetric matrix''' is a square matrix that equals its own matrix transpose. Explicitly, a sq..."

New page

{{square matrix property}}

==Definition==

===Verbal definition===

A '''symmetric matrix''' is a [[square matrix]] that equals its own [[matrix transpose]]. Explicitly, a square matrix <math>A</math> is termed a symmetric matrix if <math>A = A^T</math>.

===Algebraic 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> matrix. We say that <math>A</math> is symmetric if the following holds:

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

==Encoding==

We can encode a symmetric matrix by simply storing the entries on or above the diagonal. This is a total of <math>n(n+1)/2</math> entries instead of the total of <math>n^2</math> entries.

==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 || || || {{intermediate notions short|symmetric matrix|scalar matrix}}
|-
| [[Weaker than::diagonal matrix]] || all off-diagonal entries are zero || || || {{intermediate notions short|symmetric matrix|diagonal matrix}}
|-
| [[Weaker than::bisymmetric matrix]] || both symmetric and [[centrosymmetric matrix|centrosymmetric]] || || || {{intermediate notions short|symmetric matrix|bisymmetric matrix}}
|-
| [[Weaker than::symmetric Toeplitz matrix]] || || || || {{intermediate notions short|symmetric matrix|symmetric Toeplitz matrix}}
|}

===Other symmetry notions===

* [[Persymmetric matrix]]
* [[Centrosymmetric matrix]]
* [[Bisymmetric matrix]]