MA502 – Homework 10.

Write down detailed proofs of every statement you make

1. Let A be a real n × n matrix with an eigenvalue λ having algebraic

multiplicity n. Prove that for any t real one has

e

tA = e

λt

I + (A − λI)t + … +

(A − λI)

n−1

(n − 1)! t

n−1

!

2. Let A denote the matrix

A =

1 3

3 1 !

• Find an orthogonal matrix O such that OTAO is diagonal

• Compute the matrix e

A.

3. Consider the vector space of polynomials with real coefficients and with

inner product

hf, gi := Z 1

−1

f(t)g(t)(1 − t

2

)dt.

Apply the Graham-Schmidt process to find an orthonormal basis, with

respect to this inner product, for the subspace generated by {

√

3

2

,

√

15

2

t, t2}.

4. Let A be a real n × n matrix. Define hx, yi := Pn

i,j=1 aijxiyj

. Find

necessary and sufficient conditions on A for this operation to be a inner

product on R

3

.

5. Show that the system Ax = b has no solution and find the least square

solution of the problem Ax ≈ b with

A =

2 0

−1 1

0 2 !

and b =

1

0

−1

!

1

## Reviews

There are no reviews yet.