Question

Find the polar decomposition M=UP of the matrix:
(3 1+i)
(1-i 5)

Answer 1

To find the polar decomposition of a matrix, we need to find the eigenvalues and eigenvectors of the matrix, and then form a unitary matrix and a positive semidefinite Hermitian matrix using these eigenvalues and eigenvectors.

Step-by-step explanation:

To find the polar decomposition of the given matrix M, we need to find the unitary matrix U and the positive semidefinite Hermitian matrix P.

Step 1: Find the eigenvalues and eigenvectors of M. We can do this by solving the characteristic equation det(M - λI) = 0, where I is the identity matrix.

Step 2: The eigenvectors of M will form the columns of U and the eigenvalues will form the diagonal elements of P.

Step 3: Normalize the columns of U to make it a unitary matrix.

Step 4: Multiply U and P to get the polar decomposition M = UP.