91Ó°ÊÓ

A matrix is said to be positive definite matrix, when it is symmetric matrix and all its eigenvalues are positive.

Since \(A\)and \(B\)are positive definite matrices, therefore\({x^T}Ax > 0\) and \({x^T}Bx > 0\) for all vectors\(x\).

This implies \({x^T}(A + B)x = {x^T}Ax + {x^T}Bx > 0\) for all \(x\), which means that \(A + B\) is a positive definite.

For example; consider\(A = \left[ \begin{array}{l}1{\rm{ 3}}\\{\rm{3 5}}\end{array} \right]\)and\(B = \left[ \begin{array}{l}1{\rm{ }}2\\2{\rm{ }}1\end{array} \right]\)

To prove that the given matrix is positive definite, find the symmetric of the matrix:

\({A^T} = \left[ {\begin{array}{*{20}{c}}1&3\\3&5\end{array}} \right] = A\)

\({B^T} = \left[ {\begin{array}{*{20}{c}}1&2\\2&1\end{array}} \right] = B\)

Here,\({A^T} = A\)and\({B^T} = B\)which means it’s symmetric. So, the given matrix is positive definite.

Hence, the given statement is TRUE.