MATH 1080 Numerical Mathematics: Linear Algebra Homework 6 2022
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MATH 1080 Numerical Mathematics: Linear Algebra
Homework 6
2022
And the following problems:
1. Let Q1 , Q2 e Rn ×n be orthogonal. Prove that, for any x e Rn ,
|Q1 Q2x|2 = |x|2 .
2. Given
A = ╱ 0(1) 2(1) \ ,
find a nonsingular matrix X and a diagonal matrix D such that
X 一1 AX = D.
3. If Aqi = λiqi , i = 1, . . . , n, and x = aiqi, express Akx as a linear combination of
the eigenvectors qi .
4. Let A be a symmetric matrix and Aqi = λiqi , i = 1, 2, where q1 and q2 are orthonormal eigenvectors. Let x = q1 + eq2 .
xT Ax
a) Express the Rayleigh quotient r(x) = in terms of λ 1 , λ2 , and e.
b) Prove that lr(x) - λ 1 l < e2 lλ2 - λ 1 l.
c) If λ 1 > λ2 , prove that r(x) < λ 1 .
5. Let λ be an eigenvalue of A.
a) Show that 3λ3 + 2λ2 is an eigenvalue of 3A3 + 2A2 .
b) If A is similar to B, show that 3λ3 + 2λ2 is an eigenvalue of 3B3 + 2B2 .
2022-02-25