Suppose A is symmetric How are the eigenvalues of A related

Suppose A is symmetric. How are the eigenvalues of A related to its singular values? What if A is also positive definite? (Challenge) The largest singular value sigma_1 is called the spectral norm of the matrix, and is often used to measure the \"size\" of a matrix. Show that sigma_1 = max_x notequalto 0 ||Ax||/||x|| So sigma_1 measures the maximal amount A \"stretches\" vectors.

Solution

Given A is a symmetric matrix

So A^T=A

We know the property of the eigene

Values of A are the same Eugen values of

A^T so A^Tand A are same egen vvalues

suppose the Eigen values A are all

Positive then A is positive definite

Since A is symmetric then it has all Eigen

values are positive