Vipul: Created page with "{{square matrix property}} ==Definition== A '''bisymmetric matrix''' is a square matrix that satisfies the following equivalent conditions: # It is both a defining in..."

2014-04-29T21:22:24Z

Created page with "{{square matrix property}} ==Definition== A '''bisymmetric matrix''' is a square matrix that satisfies the following equivalent conditions: # It is both a defining in..."

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{{square matrix property}}

==Definition==

A '''bisymmetric matrix''' is a [[square matrix]] that satisfies the following equivalent conditions:

# It is both a [[defining ingredient::symmetric matrix]] and a [[defining ingredient::centrosymmetric matrix]].
# It is both a [[defining ingredient::symmetric matrix]] and a [[defining ingredient::persymmetric matrix]].
# It is both a [[defining ingredient::centrosymmetric matrix]] and a [[defining ingredient::persymmetric matrix]].

==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 Toeplitz matrix]] || || || || {{intermediate notions short|bisymmetric matrix|symmetric Toeplitz matrix}}
|-
| [[Weaker than::scalar matrix]] || || || || {{intermediate notions short|bisymmetric matrix|scalar matrix}}
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::symmetric matrix]] || symmetric about the main diagonal (top left to bottom right), or, equal to its [[matrix transpose]]|| || || {{intermediate notions short|symmetric matrix|bisymmetric matrix}}
|-
| [[Stronger than::persymmetric matrix]] || symmetric about the antidiagonal (bottom left to top right), or, commutes with the [[exchange matrix]] || || || {{intermediate notions short|persymmetric matrix|bisymmetric matrix}}
|-
| [[Stronger than::centrosymmetric matrix]] || symmetric about the center point. Note that this definition, although typically used for square matrices, also makes sense for rectangulra matrices || || || {{intermediate notions short|centrosymmetric matrix|bisymmetric matrix}}
|}

Bisymmetric matrix - Revision history

Vipul: Created page with "{{square matrix property}} ==Definition== A '''bisymmetric matrix''' is a square matrix that satisfies the following equivalent conditions: # It is both a defining in..."