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MA502 – Homework 10

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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

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