Homework 13 Solution

Description

formation geometrically (in terms of stretch factors and stretch directions)?

2. Construct an example for each of the following, or explain why such an example does not exist:

2. 1. a 2 2 matrix that is invertible but not diagonalizable

2. 2. a 2 2 non-diagonal matrix that is diagonalizable but not invertible

2. (Strang x5.2 #11) If all eigenvalues of A are 1, 1, and 2, which of the following are certain to be true? Give a reason if true or a counterexample if false.

2. 1. A is invertible.

2. 2. A is diagonalizable.

2. 3. A is not diagonalizable.

2 3

1

4. (Strang x5.2 #12) Suppose the only eigenvectors of A are multiples of ~x = 405, which of the following are

0

certain to be true? Give a reason if true or a counterexample if false.

1. 1. A is not invertible.

1. 2. A has a repeated eigenvalue.

1. 3. A is not diagonalizable.

5. (Strang x5.1 #18) Suppose a 3 3 matrix A has eigenvalues 0, 3, and 5 with associated eigenvectors ~u, ~v, and w~ respectively.

(a)	Since the eigenvalues of A are all distinct, the set f~u; ~v; w~g is			.
(b)	Write down a basis for the nullspace N(A) and the column space C(A).
(c)	Find one particular solution to A~x = ~v + w~. Find all solutions to A~x = ~v + w~.
(d)	Explain why A~x = ~u does not have a solution. (Hint: If there is a solution, then					is in C(A).
	Explain why that is impossible.)

1. 5. Is A invertible? Why or why not?

6. Let A be an n-by-n matrix. Suppose A~u = 2~u and A~v = 5~v for nonzero vectors ~u and ~v. Complete the following proof that f~u; ~vg is linearly independent.

Proof: In order for f~u; ~vg to be linearly independent, we need to show that

the only solution to the vector equation	~	is the trivial one.
	x~u + y~v = 0

Note that in this equation, ~u and ~v are known vectors, while x and y are unknown scalars. Now assume that x = a and y = b is some solution to the above equation. That is

~	(1)
a~u + b~v = 0:

MATH 141: Linear Analysis I Homework 13

Multiply both sides of equation (1) by matrix A from the left. Show your calculation details to explain why the following has to be true as well

~	(2)
2a~u + 5b~v = 0:

Explain in detail how combing vectors equations (1) and (2) leads to the conclusion that a = 0 = b. There-

fore, the only solution to x~u + y~v = ^~0 is the trivial solution.

7. In one of the reflection questions, you saw that if A is 2 2 matrix then the product of its eigenvalues is equal to det(A) and the sum is equal to trace(A). The following shows that both claims still hold true for n n matrices.

Suppose that ₁; ₂; : : : ; _n are the n eigenvalues of an n n matrix A. _i’s are the roots of the polynomial det(A I), which means that we have a factorization

det(A I) = ( 1 )( 2 ) ( n )

(3)

1. (Strang x5.1 #8) By making a clever choice of the value for in equation (3), show that det(A) is equal to the product of eigenvalues.

2. (Strang x5.1 #9) Show that trace(A) is equal to the sum of eigenvalues in three steps. First, find the

coefficient of ( )^{n 1} on the righthand side of equation (3). Next, find all terms on the righthand side

of the following that involves (	)n 1	2										3
			a21		a22					a2n
		6	a11		a12					a1n		7
det(A	I) = det		...			...				...			;
		6	a		a	n2			a	nn		7
		6	n1									7
		4										5

compare the coefficient of ( )n 1 found in these two different ways.