27. Let A be square n × n-matrix. Show that A + AT is symmetric. Show thatA+x. (^ + ¹² ) x(²2X. Ax = x.for all x in R". Conclude that x Ax ≥ 0 for all x in R" if and only if the symmetricmatrix A + AT is positive semidefinite.

Given is a square matrix of order .To show that matrix is a symmetric matrix.And to show for all…

Answered: 27. Let A be square n x n-matrix. Show…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

27. Let A be square n × n-matrix. Show that A + AT is symmetric. Show that
A
+x. (^ + ¹² ) x
(²
2
X. Ax = x.
for all x in R". Conclude that x Ax ≥ 0 for all x in R" if and only if the symmetric
matrix A + AT is positive semidefinite.

Expert Solution

Step 1: Definition of positive semidefinite matrix

Given $A$ is a square matrix of order $n cross times n$ .

To show that matrix $A plus A to the power of T$ is a symmetric matrix.

And to show $X times A X equals X times open parentheses fraction numerator A plus A to the power of T over denominator 2 end fraction close parentheses X$ for all $X element of straight real numbers to the power of n$ .

Also to conclude that $X times A X greater or equal than 0$ for all $X element of straight real numbers to the power of n$ if and only if the symmetric matrix $A plus A to the power of T$ is positive semi definite.

A matrix $M$ is called a positive semi definite matrix if and only if $M$ is symmetric matrix and $v to the power of T M v greater or equal than 0$ for all $v element of V$ .