Mathematics 244 – Linear Algebra, section 1 Problem Set 10 – Due: Wednesday, May 2, no later than 5:00pm

I. For each of the following symmetric matrices of A,

(i) Determine the eigenvalues (ii) Find an orthonormal basis of R^{n} consisting of eigenvectors of the matrix (iii) Give the spectral decomposition of A

A)

A = ( \begin{matrix}3 & 5 \\ 5 & 3 \end{matrix} )

B)

A = ( \begin{matrix}16 & −4 \\ −4 & 1 \end{matrix} )

C)

A = ( \begin{matrix}3 & 1 & 1 \\ 1 & 3 & 1 \\ 1 & 1 & 3 \end{matrix} )

II. Let A ∈ M_{n×n}(R), with entries a_{ij}, and let p(λ) = det(A − λI) be its characteristic polynomial. A) Show by induction on n that

p(λ) = (a_{11} − λ)(a_{22} − λ) · · · · · (a_{nn} − λ) + (terms of degree ≤ n − 2)

(where degree means degree as a polynomial in the variable λ). B) Deduce from part A that the coefficient of λ^{n−1} in p(λ) is (−1)^{n−1}(a_{11}+a_{22}+⋯+a_{nn}).

This scalar, the sum of the diagonal entries in A, is called the trace of A and written Tr(A).

C) Show that if A has n real eigenvalues λ_{1}, …, λ_{n} (not necessarily distinct), then Tr(A) = λ_{1} + ⋯ + λ_{n} and det(A) = λ_{1} · · · · · λ_{n}.

III. An n × n matrix Q is said to be orthogonal if Q is invertible and Q^{−1} = Q^{t}. A) Show that if Q is orthogonal, then the columns of Q form an orthonormal basis of

R^{n}. B) Show conversely that if the columns of Q form an orthonormal basis for R^{n}, then Q

is orthogonal. C) Show that if P and Q are orthogonal, then P Q is also orthogonal. D) Show that if Q is orthogonal, then Q^{−1} is also orthogonal.