Matrix analysis and applied linear algebra

hw9.pdf

MATH 92.564 - Spring 2015

Homework 9 (due on April 14)

1. Find all eigenvalues and eigenvectors of A =

 7 −20 23 −7 −1

3 −10 2

 .

2. Let u be a unit vector in Rn. The orthogonal projection onto {u}⊥ is given by P = I−uuT .

(a) Prove that P is a singular matrix.

Hint: A matrix P is singular if the equation Px = has a nonzero solution.

(b) Prove that rank(P ) = n− 1. Hint: use the formula n = rank(P ) + dim N(P ).

3. Let x = 1

  1−2 −2

 . Extend x to an orthonormal basis for R3, that means, find two unit

vectors u, v such that {x, u, v} is an orthonormal basis of R3. Hint: use an elementary reflector.

4. Define the following determinants

D1 = det ( 2 )

= 2

D2 = det

( 2 −1 −1 2

) = 3

D3 = det

  2 −1 0−1 2 −1

0 −1 2

 

D4 = det

 

2 −1 0 0 −1 2 −1 0

0 −1 2 −1 0 0 −1 2

 

...

Dn = det

 

2 −1 0 · · · 0 −1 2 −1 · · · 0

. . . . . .

. . .

0 · · · −1 2 −1 0 · · · 0 −1 2

 

n×n

(a) Use a cofactor expansion to derive the formula Dn = 2Dn−1 −Dn−2 (n ≥ 3). (b) Use the above formula to compute D3, D4, D5.