Chapter 5 – Key Concepts

Question 1

What is an eigenvector of an n×n matrix A? An eigenvalue?

Answer 1

An eigenvector of an n×n matrix A is a nonzero vector x such that Ax = λx for some scalar λ. A scalar λ is called an eigenvalue of A if there is a nontrivial solution x of Ax = λx; such an x is called an eigenvector corresponding to λ.

Question 2

What are the eigenvalues of a triangular matrix?

Answer 2

The eigenvalues of a triangular matrix are the entries on its main diagonal.

Answer 3

If v₁, …, v_r are eigenvectors that correspond to distinct eigenvalues λ₁, …, λ_r of an n×n matrix A, then the set {v₁, …, v_r} is linearly independent.

Question 4

What is the determinant of a matrix A in terms of its row echelon form U?

Answer 4

det A = (-1)^r ⋅ (product of pivots in U) if A is invertible, and det A = 0 when A is not invertible.

Question 5

What is the IMT continued for eigenvectors? That is, let A be n×n. Then A is invertible if and only if:

s. The number 0 is or is not an eigenvalue of A?
t. The determinant of A is or is not equal to 0?

Answer 5

s. The number 0 is not an eigenvalue of A.
t. The determinant of A is not zero.

Question 6

Let A and B be n×n matrices. Evaluate the following properties of determinants:

a. A is invertible ⇔ det A ≠ ?
b. det(AB) = what in terms of determinants of A and B?
c. how is det A^T related to det A?
d. If A is triangular, then det A is what product?
e. How do row replacement, interchange, and scaling affect the determinant of A?

Answer 6

a. A is invertible ⇔ det A ≠ 0
b. det(AB) = det(A) det(B)
c. det(AT) = det(A)
d. If A is triangular, det(A) = product of the main diagonal of A.
e. Row replacement doesn’t affect det(A), an interchange between A and A’ ⇒ det(A) = -det(A’), and row scaling scales the determinant by the same factor: if A’ is A with one row scaled by k, then k det(A) = det(A’)

Question 7

If λ is an eigenvalue of A, then what must be true about the characteristic equation det(A - λI)?

Answer 7

A scalar λ is an eigenvalue of an n×n matrix A if and only if λ satisfies the characteristic equation:

det(A - λI) = 0

Question 8

If A and B are similar, what must be true about their eigenvalues and characteristic polynomials?

Answer 8

If n×n matrices A and B are similar, then they have the same characteristic polynomial and hence the same eigenvalues (with the same multiplicities).

Question 9

If A and B have the same eigenvalues, are they similar? What is the relationship between row equivalent and similarity?

Answer 9

Matrices are not necessarily similar if they have the same eigenvalues. Similarity is not at all the same thing as row equivalence, as row operations usually alter the eigenvalues of a matrix.

Question 10

What is the diagonalization theorem?

Answer 10

An n×n matrix A is diagonalizable ⇔ A has n linearly independent eigenvectors.

Also, A = PDP^-1, with D a diagonal matrix ⇔ the columns of P are n linearly independent eigenvectors of A. In this case, the diagonal entries of D are eigenvalues of A that correspond, respectively, to the eigenvectors in P.

Question 11

Let A be an n×n matrix whose distinct eigenvalues include λ₁, …, λ_p. Evaluate the following:

a. For 1 ≤ k ≤ p, the dimension of the eigenspace for λ_k ≤ what?
b. The matrix A is diagonalizable ⇔ the sum of the dimensions of the eigenspaces equals what? And this happens ⇔ (i) what about the factorization of the characteristic polynomial? ⇔ (ii) the dimension of the eigenspace for each λ_k equals what?
c. If A is diagonalizable and B_k is a basis for the eigenspace corresponding to λ_k ∀ k, then the collection of vectors in the sets B₁, …, B_p forms an eigenvector basis for what?

Answer 11

a. The multiplicity of the eigenvalue λ_k.
b. n. (i) it factors completely into linear factors. (ii) the multiplicity of λ_k.
c. ℝⁿ

Question 12

Let T : V → W, where V ⊆ ℝⁿ and W ⊆ ℝ^m. Let B be a basis for V, and C be a basis for W. What is the matrix equation expression for this mapping in terms of a vector x ∈ V?

Answer 12

[T(x)]_C = M [x]_B where M is the matrix for T relative to the bases B and C.

Question 13

If [T(x)]_C = M [x]_B, what are columns of the matrix M in terms of b₁, …, b_n ∈ B?

Answer 13

M = [[T(b₁)]_C … [T(b_n)]_C]

Question 14

What is the Diagonal Matrix Representation Theorem?

Answer 14

Suppose A = PDP^-1, where D is a diagonal n×n matrix. If B is the basis for ℝⁿ formed from the columns of P, then D is the B-matrix for the transformation x ↦ Ax.

Question 15

Let A be a real 2×2 matrix with a complex eigenvalue λ = a + bi (b ≠ 0) and an associated eigenvector v ∈ ℂ². Then A = PCP^-1, where P is what and C is what?

Answer 15

P = [Re(v) Im(v)]

C = [[a -b] [b a] ]

Question 16

What is the power method for estimating a strictly dominant eigenvalue?

Answer 16

1. Select an initial vector x₀ whose largest entry is 1.
2. For k = 0, 1, …,
   a. Compute Ax_k.
   b. Let µ_k be an entry in Ax_k whose absolute value is as large as possible.
   c. Compute x_k+1 = (1/µ_k) Ax_k.
3. For almost all choices of x₀, the sequence {µ_k} approaches the dominant eigenvalue, and the sequence {x_k} approaches a corresponding eigenvector.

Question 17

What is the inverse power method for estimating an eigenvalue λ of A?

Answer 17

1. Select an initial estimate α sufficiently close to λ.
2. Select an initial vector x₀ whose largest entry is 1.
3. For k = 0, 1, …,
   a. Solve (A - αI)y_k = x_k for y_k.
   b. Let µ_k be an entry in y_k whose absolute value is as large as possible.
   c. Compute v_k = α + (1/µ_k).
   d. Compute x_k+1 = (1/µ_k)y_k.
4. For almost all choices of x₀, the sequence {v_k} approaches the eigenvalue λ of A, and the sequence {x_k} approaches a corresponding eigenvector.