91Ó°ÊÓ

A matrix M is positive-definite if and only if any of the equivalent requirements are met. A diagonal matrix with positive real elements is equivalent with M. All of M's eigenvalues are real and positive, and it is symmetric or Hermitian.

Symmetric matrix is positive definite all of its eigen values are positive. Consider ${\mathrm{A}}_{\mathrm{n}×\mathrm{n}}$to be a positive definite matrix is represented as

$\mathrm{A}={\left[\begin{array}{cccccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& ...& {\mathrm{a}}_{1\mathrm{i}}& ...& {\mathrm{a}}_{1\mathrm{n}}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}& ...& {\mathrm{a}}_{2\mathrm{i}}& ...& {\mathrm{a}}_{2\mathrm{n}}\\ \mathrm{M}& \mathrm{M}& ...& \mathrm{M}& ...& \mathrm{M}\\ {\mathrm{a}}_{\mathrm{i}1}& {\mathrm{a}}_{\mathrm{i}2}& ...& {\mathrm{a}}_{\mathrm{ii}}& ...& {\mathrm{a}}_{\mathrm{in}}\\ \mathrm{M}& \mathrm{M}& ...& \mathrm{M}& ...& \mathrm{M}\\ {\mathrm{a}}_{\mathrm{n}1}& {\mathrm{a}}_{\mathrm{n}2}& ...& {\mathrm{a}}_{\mathrm{ni}}& ...& {\mathrm{a}}_{\mathrm{nn}}\end{array}\right]}_{\mathrm{n}×\mathrm{n}}$

Here ${\mathrm{a}}_{\mathrm{ij}}$represents the diagonal entries of the matrix A. Consider the standard basis are represented as

${\mathrm{e}}_{\mathrm{i}}={\left[\begin{array}{c}0\\ 0\\ .\\ 1\\ .\\ 0\end{array}\right]}_{\mathrm{n}×1}$

Consider the quadratic form of A as

role="math" localid="1659670850035" $$\begin{gathered} \mathrm{q}\left({\mathrm{e}}_{\mathrm{i}}\right)={\mathrm{e}}_{\mathrm{i}}^{\mathrm{T}}{\mathrm{Ae}}_{\mathrm{i}}(\mathrm{where}\mathrm{i}=1,...,\mathrm{n}) \\ \mathrm{q}\left({\mathrm{e}}_{\mathrm{i}}\right)â‰\yen 0\mathrm{As}{\mathrm{A}}_{\mathrm{n}×\mathrm{n}}\mathrm{is}\mathrm{a}\mathrm{positive}\mathrm{definite}\mathrm{matrix}.\mathrm{The}\mathrm{equation}\mathrm{becomes} \end{gathered}$$

$\mathrm{q}\left({\mathrm{e}}_{\mathrm{i}}\right)={\mathrm{e}}_{\mathrm{i}}^{\mathrm{T}}{\mathrm{Ae}}_{\mathrm{i}}â‰\yen 0$

Consider as${\mathrm{e}}_{\mathrm{i}}^{\mathrm{T}}{\mathrm{Ae}}_{\mathrm{i}}â‰\yen 0$

$$\begin{gathered} {\left[\begin{array}{c}0\\ 0\\ .\\ 1\\ .\\ 0\end{array}\right]}_{\mathrm{n}×1}^{\mathrm{T}}{\left[\begin{array}{cccccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& ...& {\mathrm{a}}_{1\mathrm{i}}& ...& {\mathrm{a}}_{1\mathrm{n}}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}& ...& {\mathrm{a}}_{2\mathrm{i}}& ...& {\mathrm{a}}_{2\mathrm{n}}\\ \mathrm{M}& \mathrm{M}& ...& \mathrm{M}& ...& \mathrm{M}\\ {\mathrm{a}}_{\mathrm{i}1}& {\mathrm{a}}_{\mathrm{i}2}& ...& {\mathrm{a}}_{\mathrm{ii}}& ...& {\mathrm{a}}_{\mathrm{in}}\\ \mathrm{M}& \mathrm{M}& ...& \mathrm{M}& ...& \mathrm{M}\\ {\mathrm{a}}_{\mathrm{n}1}& {\mathrm{a}}_{\mathrm{n}2}& ...& {\mathrm{a}}_{\mathrm{ni}}& ...& {\mathrm{a}}_{\mathrm{nn}}\end{array}\right]}_{\mathrm{n}×\mathrm{n}}{\left[\begin{array}{c}0\\ 0\\ .\\ 1\\ .\\ 0\end{array}\right]}_{\mathrm{n}×1}â‰\yen 0 \\ {\left[00...1...0\right]}_{\mathrm{n}×1}{\left[\begin{array}{cccccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& ...& {\mathrm{a}}_{1\mathrm{i}}& ...& {\mathrm{a}}_{1\mathrm{n}}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}& ...& {\mathrm{a}}_{2\mathrm{i}}& ...& {\mathrm{a}}_{2\mathrm{n}}\\ \mathrm{M}& \mathrm{M}& ...& \mathrm{M}& ...& \mathrm{M}\\ {\mathrm{a}}_{\mathrm{i}1}& {\mathrm{a}}_{\mathrm{i}2}& ...& {\mathrm{a}}_{\mathrm{ii}}& ...& {\mathrm{a}}_{\mathrm{in}}\\ \mathrm{M}& \mathrm{M}& ...& \mathrm{M}& ...& \mathrm{M}\\ {\mathrm{a}}_{\mathrm{n}1}& {\mathrm{a}}_{\mathrm{n}2}& ...& {\mathrm{a}}_{\mathrm{ni}}& ...& {\mathrm{a}}_{\mathrm{nn}}\end{array}\right]}_{\mathrm{n}×\mathrm{n}}{\left[\begin{array}{c}0\\ 0\\ .\\ 1\\ .\\ 0\end{array}\right]}_{\mathrm{n}×1}â‰\yen 0 \end{gathered}$$

$$\begin{gathered} {\left[{\mathrm{a}}_{\mathrm{i}1}{\mathrm{a}}_{\mathrm{i}2}...{\mathrm{a}}_{\mathrm{ii}}...{\mathrm{a}}_{\mathrm{in}}\right]}_1{\left[\begin{array}{c}0\\ 0\\ ..\\ 1\\ ..\\ 0\end{array}\right]}_{\mathrm{n}×1}â‰\yen 0 \\ {\mathrm{a}}_{\mathrm{i}1}×0+{\mathrm{a}}_{\mathrm{i}2}×0+...{\mathrm{a}}_{\mathrm{ii}}×1+...+{\mathrm{a}}_{\mathrm{in}}×0â‰\yen 0 \\ 0+0+...+{\mathrm{a}}_{\mathrm{ii}}+...+0â‰\yen 0 \\ {\mathrm{a}}_{\mathrm{ii}}â‰\yen 0 \end{gathered}$$

Therefore, we can conclude that the diagonal elements of a positive definite matrix are positive.