Answer to Question #47550 in Matrix | Tensor Analysis for Andy Ladener

Question

1. Find the matrix that represents the linear transformation of the plane obtained by reflecting in the line y=x, then rotating anticlockwise through an angle of 45 degrees, and finally reflecting in the y-axis. Give a simpler geometrical description of what this transformation does.

Factorize the denominator of f(x)=(2x^5+15x^4+15x^3+2x^2+2)/(x^5+2x^4+x^3-x^2-2x-1) completely, and use this to write f(x) as a partial fraction.

Find the eigenvalues and eigenvectors of the symmetric matrix S = [[-1,2,0],[2,2,1],[0,1,-1]]. Let M be the 3x3 matrix whose columns are the eigenvectors you have found. Evaluate M^T*M, with as little computation as possible. Give reasons for any computations you were able to omit.