Introduction to Computational Linear Algebra

52 Introduction to Computational Linear Algebra

Exercise 2.5

Prove, for U ∈ C

n×n

unitary, that |det(U)| = 1.

Exercise 2.6

Prove, for Q ∈ R

n×n

orthogonal, that det(Q) = ±1.

Exercise 2.7

Prove, for A ∈ C

n×n

Hermitian, that Λ(A) ⊂ R.

Exercise 2.8

Show, directly and through using Schur’s decomposition, that for any n by n

matrix, p

A

(λ) is a polynomial of degree exactly n.

Exercise 2.9

Show if A ∈ R

n×n

is spd, then Λ(A) ⊂ R

+

.

Exercise 2.10

Show if A ∈ R

n×n

is spsd, then Λ(A) ⊂ R

+

∪ {0}.

Exercise 2.11

Prove that for A ∈ R

n×n

:

A pd ⇐⇒

1

2

(A + A

T

) spd

Exercise 2.12

Let A ∈ R

m×n

be a general rectangular matrix (m < n, m = n or m > n).

Prove that the matrices AA

T

∈ R

m,m

and A

T

A