Description
Problem 1:
Find the diagonalization = Λ −1 of the following matrix:
-
3
0
0
0
= [
0
−1
3
1]
−2
0
4
0
2
−2
1
2
Problem 2:
Find the Schur factorization = for the following matrix.
(Hint: Follow the proof of existence of the factorization.)
-
4
−2
1
= [−2
4
2]
1
1
4
Problem 3:
Calculate the Rayleigh quotients = ( ) for the following matrix and given vectors . How far is each from the closest eigenvalue of ?
-
4
6
1
= [6
4
6],
1
6
4
1.5
1
1
1
1=[2
] ,
2 = [2.1] ,
3=[ 0
] ,
4=[1]
1
1
−1.1
1
Problem 4:
Let be a symmetric matrix and let 1 ≤ 2 ≤ ⋯ ≤ be its eigenvalues. Show that for any ≠ 0 the Rayleigh quotient ( ) = obeys 1 = min ( ) and = max ( ).
x≠0 x≠0
(Hint: Use orthogonal diagonalization of the matrix A.)
