** 2301336 LINEAR ALGEBRA II PRACTICE FINAL EXAM FEBRUARY 2014 **
For this exam, the standard inner product onCnis given by~x·~y=~xH~yfor all~x, ~y∈Cn. 1. LetB={e−t,sint,cost}be an ordered basis forH = SpanB.
Define a linear transformationT :H→HbyT(f) =f′.
(1a) Find[T]B. (1b) Is[T]B unitarily diagonalizable? Explain.
2. LetA= 1
√2
−1 i i 1
. Check ifAis
(2a) symmetric (2b) Hermitian (2c) unitary (2d) normal.
3. Suppose the characteristic polynomial of a3×3matrixAisx(x−1)(x+ 1).
(3a) IsAinvertible? Explain. (3b) Find the characteristic polynomial ofA2. (3c) Show thatA3 =A.
4. LetAandB be Hermitian matrices. Prove thatABis Hermitian if and only ifAB=BA.
5. Unitarily diagonalize the Hermitian matrixA=
3 1−i 1 +i 2
.
6. LetAbe any complex matrix. Show that
(6a)AHAis Hermitian, and (6b) the eigenvalues ofAHAare all non-negative.
7. Let
A=
−1 1 1 −1 0 −1 0 0 0 0 −1 1
0 0 0 −1
.
Write down the Jordan form forAand find its minimal polynomial.
8. LetJ =
5 1
5 5
−2 1
−2
be in Jordan form.
(8a) How many Jordan blocks doesJ have? (8b) Find the characteristic polynomial ofJ. (8c) Find the minimal polynomial ofJ. (8d) ComputeJsfor any positive integers.
9. Determine (with short reason or counter example) whether the following statements are TRUE or FALSE.
(9a) IfA∈Mn(C)is a Hermitian matrix, thendetAis a real number.
(9b) IfU is ann×nunitary matrix, thenU+ 2Inis invertible.
(9c)
4 0 0 4
and
4 1 0 4
are similar.
10. Exercises V –15, 18, 19, 21, 26, 30, 31, 34, 36.
