Chapter 4

Question 1

Let V be a vector space over F and T: V → V is a linear map. What is an eigenvector?

Answer 1

A non-zero vector 𝐯 is an eigenvector if T(𝐯) = ƛ𝐯

Question 2

Let V be a vector space over F and T: V → V is a linear map. What is the ƛ-eigenspace?

Answer 2

V = {𝐯 ⋲ V: T(𝐯) = ƛ𝐯}

= {𝐯 ⋲ V: (T - ƛI)𝐯 = 𝟎}

= ker (T - ƛI)

Question 3

Let V be a vector space over F and T: V → V is a linear map. How do you know the ƛ-eigenspace is a subspace of V?

Answer 3

Since the eigenspace is a kernel

Question 4

Why do we take 𝐯 ⋲ ℂ^n and ƛ ⋲ ℂ, even if the matrix A has real components?

Answer 4

A real matrix might have complex eigenvectors

as real polynomials can have complex roots

Question 5

What is the characteristic polynomial of an n x n matrix A?

Answer 5

𝓧_A (t) = det (A - tI)

Question 6

How are the eigenvalues of A and the roots of characteristic polynomial related?

Answer 6

The eigenvalues of A are the roots of the characteristic polynomial

Question 7

What are the eigenvalues of an upper triangular matrix?

Answer 7

Its diagonal entries

Question 8

What is true of the determinant and characteristic polynomial for similar matrices?

Answer 8

They are the same

Question 9

If A and B are similar, so that A = P⁻¹BP, what is 𝓧A(t) equal to?

Answer 9

𝓧A(t) = 𝓧B(t)

Question 10

Suppose A is an n x n matrix and ƛ ⋲ ℂ is an eigenvalue of A. What is the geometric multiplicity of ƛ?

Answer 10

The dimension of the ƛ-eigenspace

Question 11

Suppose A is an n x n matrix and ƛ ⋲ ℂ is an eigenvalue of A. What is the algebraic multiplicity of ƛ?

Answer 11

The multiplicity of the root ƛ in the characteristic polynomial

Question 12

How are the geometric and algebraic multiplicities of ƛ related?

Answer 12

The geometric multiplicity of ƛ is less than or equal to the algebraic multiplicity of ƛ

Question 13

If 𝐯₁, …, 𝐯n are eigenvectors of A corresponding to distinct eigenvalues ƛ₁, …, ƛn, then what does this mean for {𝐯₁, …, 𝐯n}?

Answer 13

{𝐯₁, …, 𝐯n} is linearly independent

Question 14

When is an n x n matrix A called diagonalisable?

Answer 14

If it is similar to a diagonal matrix

That is, there exists a P, such that P⁻¹AP is diagonal

Question 15

Let A be an n x n matrix and let V = ℂ^n. What statements about A are equivalent?

Answer 15

1. A is diagonalisable
2. A has n linearly independent eigenvectors
3. V = V_ƛ₁ ⊕ … ⊕ V_ƛk, where ƛ₁, …, ƛk are the distinct eigenvalues of A
4. For all eigenvalues ƛ, the geometric and algebraic multiplicities are equal

Question 16

What is a practical way to check if a matrix is diagonalisable or not?

Answer 16

See if A has n linearly independent eigenvectors

Question 17

A is an n x n matrix, and its characteristic polynomial has n distinct roots in ℂ. Is A diagonalisable?

Answer 17

Yes

Question 18

If A is an n x n matrix over ℂ, and

f(t) = a₀ + a₁t + … a_m t^m is a polynomial with complex coefficients. How is f(A) defined?

Answer 18

f(A) = a₀ + a₁A + … + a_m A^m

which is an n x n matrix

Question 19

What is the Cayley-Hamilton theorem?

Answer 19

Every square matrix A satisfies its own characteristic equation, 𝓧A(A) = 0

Question 20

What is a Jordan block matrix? (try to draw)

Answer 20

A square matrix of the form

J_n(ƛ) = (ƛ 1 0 … 0)
( … … 1)
(0 … … 0 ƛ)

with ƛ on the diagonal and 1 just above the diagonal (for some fixed scalar ƛ). The subscript n gives the size.

Question 21

When is a square matrix A said to be in Jordan Normal Form (JNF)?

Answer 21

If it consists of Jordan block matrices on the diagonal, with zeros elsewhere

Question 22

Is a Jordan block matrix in JNF?

Answer 22

Yes

Question 23

What is any n x n matrix similar to?

Answer 23

A matrix in Jordan normal form

Question 24

What is the determinant of a matrix equal to in terms of its eigenvalues?

Answer 24

The product of its eigenvalues

Question 25

Let A be an n x n matrix which is similar to a JNF matrix B. What is true for each eigenvalue ƛ?

Answer 25

1) The algebraic multiplicity of ƛ is the total number of copies of ƛ on the diagonal of B
2) The geometric multiplicity of ƛ is the number of Jordan blocks with eigenvalue ƛ in B