Positive Matrix proof (1 Viewer)

InteGrand · Mar 8, 2017

integral95 said:
$a) Prove that a Square Matrix $ \ G \ $ is a positive definite if and only if $ \ G+G^T \ $ is a positive definite$

$b) Prove that an $ \ n \ times \ n \ $ symmetric matrix $ \ G \ $ is a positive definite if and only if $ \ G^{-1} \ $ is positive definite$

Here's some hints.

$\noindent a) Show $\mathbf{x}^{T}G^{T}\mathbf{x} =\mathbf{x}^{T}G\mathbf{x}$ for all $\mathbf{x}\in \mathbb{R}^{n}$ (where $n$ is the matrix size), e.g. by considering properties of transpose / inner products or whatever method you like.$

$\noindent b) One way is to consider eigenvalues (recall the eigenvalues of $G^{-1}$ are precisely the inverses (reciprocals) of the eigenvalues of $G$).$

integral95 · Mar 8, 2017

InteGrand said:
Here's some hints.

$\noindent a) Show $\mathbf{x}^{T}G^{T}\mathbf{x} =\mathbf{x}^{T}G\mathbf{x}$ for all $\mathbf{x}\in \mathbb{R}^{n}$ (where $n$ is the matrix size), e.g. by considering properties of transpose / inner products or whatever method you like.$

$\noindent b) One way is to consider eigenvalues (recall the eigenvakues of $G^{-1}$ are precisely the inverses (reciprocals) of the eigenvalues of $G$).$

Sweet I know how b) works now, I'm not sure which transpose properties to consider for a), having a mind blank.

InteGrand · Mar 8, 2017

integral95 said:
Sweet I know how b) works now, I'm not sure which transpose properties to consider for a), having a mind blank.

$\noindent Make use of the property $\left(\mathbf{u}^{T}M\mathbf{v}\right)^{T} = \mathbf{v}^{T}M^{T}\mathbf{u}$.$

Positive Matrix proof (1 Viewer)

integral95

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InteGrand

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integral95

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InteGrand

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