Calculate its characteristic polynomial and its eigenvalues

Question

MIDTERM 2-

Problem 1- Consider the square matrix

(a) Calculate its characteristic polynomial and its eigenvalues.

(b) Find the eigenvectors of A.

(c) Diagonalize the matrix A.

Problem 2- (a) Define an inner product in a vector space V.

(b) For p, q ∈ P₂(R) (polynomials up to degree 2 in one variable, with real coefficients) define

(p, q) = p(-1)q(-1) + p(0)q(0) + p(1)q(1) .

Show that this is an inner product.

(c) Find an orthogonal basis in P₂(R) with respect to the above inner product.

Problem 3- Find a least squares solution xˆ for the linear system Ax = b, where

Problem 4- Mark each statement True or False. Justify your answers.

(a) If A⁴ = A then -1 is not an eigenvalue for A.

(b) An invertible matrix has only nonzero eigenvalues.

Problem 5- A 3 × 3 symmetric matrix A has eigenvalues λ₁ = 0, λ₂ = 1, and λ₃ = 2. The first two eigenvectors are

(a) Find the third eigenvector v³.

(b) Find the matrix A.

Problem 6- Prove the following inequalities for vectors in an inner product space V:

(a) For any two vectors u, v we have

(u, v) 2 ≤ ||u||² · ||v||²

(b) If v₁, . . . , v_k is an orthonormal set, then for each vector x we have

||x||² ≥  (x, v₁)² + . . . + (x, v^k)².