Symmetric positive-definite matrix - Revision history

Vipul: Created page with "{{real square matrix property}} ==Definition== Suppose $n$ is a positive integer and $A$ is a $n \times n$ square matrix. We say that
2014-05-01T16:47:34Z

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..."

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{{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}}
|}