TRUE OR FALSE (a) If U is unitary, then U is itself unitarily diagonalizable. This means there is a unitary V so that U = V DV H where D is diagonal. (b) For any diagonalizable matrix A, one can use Gram-Schmidt to find an orthogonal basis consisting of eigenvectors. (c)The collection of rank k n × n matrices is a subspace of Rn×n, for k < n. (d) If A is unitary, then |λ| = 1 for all eigenvalues λ of A. (e) If p(t) is a polynomial and v
TRUE OR FALSE
(a) If U is unitary, then U is itself unitarily diagonalizable. This means there is a unitary V so that U = V DV H where D is diagonal.
(b) For any diagonalizable matrix A, one can use Gram-Schmidt to find an orthogonal basis consisting of eigenvectors.
(c)The collection of rank k n × n matrices is a subspace of Rn×n, for k < n.
(d) If A is unitary, then |λ| = 1 for all eigenvalues λ of A.
(e) If p(t) is a polynomial and v is an eigenvector of A with associated eigenvalue λ, then p(A)v = p(λ)v.
(f) If A and B are both n n and is a basis for Cn consisting of eigenvectors for both A and B, then A and B commute.
(g) Any matrix A can be written as a weighted sum of rank 1 matrices.
(h)For all Hermitian matrices A, there is a matrix B so that BHB = A.
(i) There are linear maps L : R5 → R4 such that dim(ker(L)) = 2 = dim(rng(L)).
(j) If A is invertible, then ABA−1 = B.
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