Linear Algebra Answers

Prove that, if S is square, U is unitary, and U^(conjugate transpose)SU = T is upper triangular, then the eigenvalues of S and T are the same and S and T have the same trace. (Use the facts that det(AB) = det(A)det(B), and tr(ABC) = tr(CAB) = tr(BCA)).

Prove that the eigenvalues of an upper triangular matrix T are the entries of its main diagonal, so that the trace of T is the sum of its eigenvalues.

Shur's Lemma: For any square matrix S therer is a unitary matrix U such that
U^(conjugate transpose)SU = T is an upper triangular matrix.

Use Shur's Lemma to prove that
(a) If S^(conjugate transpose)=S then there is a unitary matrix U such that U^(conjugate transpose)SU is a real diagonal matrix.

(b) If S is real and and S^T = S then there is an orthogonal matrix O such that O^TSO is a real diagonal matrix.

Associated with each equation (Ax)_i=b_i in the system Ax=b there is a hyperplane H_i defined to be the subset of J-dimensional column vectors given by:

H_i = {x|(Ax)_i=b_i}.

Show that the ith column of A^(conjugate transpose) is normal to the hyperplane H_i; that is, it is orthogonal to every vector lying in H_i.

Ok so I have 3 matrix' and its A B C and I have to solve (-1 means inverse) for (A-1xB)-1xC so does (A inverse times B) the whole thing inverse times C = (B-1A)C?

1 a) Show that if A is nonsingular symmetric matrix, then A^-1 is also symmetric. Write your
justication in clear sentences.
b) An n x n matrix A is called skew-symmetric if A = -A^T . Show that if n is odd a skew-symmetric matrix is singular.

1.a) Prove that the product A = v(w^T) of a nonzero m x 1 column vector v by a nonzero 1 x n
row vector w^T is an m x n matrix of rank 1. [Hint: do a few small examples]
b) Now show that if A is an m x n matrix of rank 1, then there exist a nonzero m x 1 column
vector v and a nonzero 1 x n row vector w^T such that A = v(w^T).

Is orthogonality reflexive, symmetric, and transitive? If so, it is an equivalence relation. If not true, find a counter-example.

1. (2 Points) Explain clearly why the solution to the homogeneous system Ax = 0 with a nonsingular
coecient matrix is x = 0.

2. (2 Points) Under what conditions does a diagonal matrix D = diag(d1,d2,.....,dn) have an inverse
D^-1? What is the inverse D^-1 when these conditions are met? Justify your answers.

3. (3 Points) Let A be an mxn matrix and B be an nxm matrix where m > n. Show that the nxn matrix AB is not invertible.

1. (2 Points) Explain clearly why the solution to the homogeneous system Ax = 0 with a nonsingular
coecient matrix is x = 0.

2. (2 Points) Under what conditions does a diagonal matrix D = diag(d1,d2,.....,dn) have an inverse
D^-1? What is the inverse D^-1 when these conditions are met? Justify your answers.

3. (3 Points) Let A be an mxn matrix and B be an nxm matrix where m > n. Show that the nxn matrix AB is not invertible.