Let B = [201|020|102] (3x3 matrix)

Question

(a) What does the Spectral Theorem guarantee about the matrix B?

(b) Find the eigenvalues and eigenvectors of B.

(c) Assume that f(x,y,z) = x2 +y2 +z2 +xz has a critical point at (0,0,0). Is f(0,0,0) a maximum, minimum, saddle point, or do we not have enough information to determine this?

Answer 1

a) This is a symmetric matrix, so the spectral theorem says there are 3 real eigenvalues (possibly not all distinct, though here they are) and their associated eigenvectors can be chosen to form an orthornormal basis.

b) The eigenvalues and their corresponding eigenvectors are:

1, [-1,0,1],

2, [0,1,0]

3, [1,0,1].

c) Do not have enough information.