Eigen Values

Question 1

What is a hermitian matrix A^*.

Answer 1

A = (aij) => A^* = (conjugate aij)^T.

So ( 1 i )
1 + i, 1 - 2i )

= ( 1 1 - i )
-i 1 + 2i )

Question 2

What does skew-symmetric / antisymmetric matrix mean?

Answer 2

A^T = -A

Question 3

State the fundamental theorem of algebra.

Answer 3

Every p(x) ∈ ℂ[x], p(x) != 0, of degree n has precisely n complex roots, when counted with their multiplicity.

Question 4

Define eigenvalue and eigenvector.

Answer 4

Let A ∈ Mn(F). A scalar λ is an eigenvalue of A is ∃ v ∈ F^n, v != 0n, s.t.

Av = λv

v is an eigenvector for the eigenvalue λ.

Question 5

Define spectrum of A and how it is denoted.

Answer 5

Set of all eigenvalues of A.

σ(A)

Question 6

T or F. If A ∈ Mn(R), then the eigenvalues of A must be strictly within R.

Answer 6

False, they can also be complex numbers or any set that R/F is a proper subset of.

Question 7

T or F. If v is a eigen-vector of A then kv is also an eigenvector of A.

Answer 7

True.

Question 8

How do we compute eigenvalues? Let A ∈ Mn(F).

Answer 8

det(A - λIn) = 0 where λ is the e-value.

Roots of characteristic polynomial.

Question 9

What is the characrteristic (a) polynomial (b) equation of A?

Answer 9

(a) Pa(x) = det(A-xIn)
(b) det(A-xIn) = 0

Question 10

Define eigenspace of A. A ∈ Mn(F).

Answer 10

The set of λ eigenvectors of A union with the zero vector.

Eλ := N(A - λIn)

Question 11

T or F. Let A ∈ Mn(F) and let λ ∈ σ(A) then Esubλ is a subspace of F^n.

Answer 11

True.

Question 12

Distinguish between algebraic multiplicity and geometric multiplicity.

Answer 12

gsubx, the geometric multiplicity of λ is the dimension of Esubλ.

asubλ, the algebraic multiplicity of λ is the multiplicity of λ as a root of the characteristic polynomial Pa(x).

Question 13

State the important result relating to asubλ and gsubλ.

Answer 13

For all A ∈ Mn(F0 and for all λ ∈ σ(A)
1 <= gsubλ <= asubλ <= n

Question 14

Define similar matrices.

Give proper notation!

Answer 14

Let A,B ∈ Mn(F). A is similar to B if ∃ an invertible matrix C ∈ Mn(F) s.t.

A = C^-1BC

A ~ B

Question 15

T or F. If A ~ B then B ~ A.

Answer 15

True.

Question 16

T or F. If A ~ B and B ~ C then A ~ C.

Answer 16

True.

Question 17

T or F. A !~ A.

Answer 17

False.

Question 18

What can we say about their determinants and traces of A ~ B?

Answer 18

det(A) = det(B)
trace(A) = trace(B)

Question 19

Define trace(A).

Answer 19

The sum of the diagonal elements of the matrix A.

Question 20

T or F. Let A ~ B, let Pa(x) be the characteristic polynomial of A and let Qa(x) be the characteristic polynomial of A. Pa(x) !- Qa(x).

Answer 20

False. They are always the same.

Question 21

T or F. If A ~ B, then their e-values are the same.

Answer 21

True.

Question 22

Let A ∈ Mn(R) or Mn(C). We have σ(A), can we calculate the trace and/or the determinant?

Answer 22

det(A) = λ1λ2…λn
trace(A) = λ1 + λ2 + … + λn

Question 23

T or F. A ∈ Mn(R) of Mn(C) is invertible if and only if 0 !∈ σ(A).

Answer 23

True.

Question 24

We know that A is an upper triangular matrix how can we get its e-values.

Answer 24

Values along diagonal are e-values.

Question 25

If k ∈ N, λ ∈ σ(A), v ∈ Esubλ \ {0}, what can we say about λ^k?

Answer 25

It is also an e-value of A^k and v ∈ Esubλ^k of A

Question 26

T or F. Non square matrices can have eigenvalues.

Answer 26

False.

Question 27

T or F. Every eigenvalue has at least one eigenvector.

Answer 27

True.

Question 28

Let p(x) ∈ R[x] be a real polynomial of degree n, what can we say about it?

Answer 28

1. If z ∈ C is a root of p(x), z conjugate is a root of p(x). 2. If n is odd, then p(x) has at least one real root. 3. p(x) can be written as a product of real polynomials whose degree is at most 2.

Question 29

T or F. All non real e-values come is complex conjugate pairs.

Answer 29

True.

Question 30

Let A be a real symmetric matrix or a complex hermitian matrix, what can we say?

Answer 30

The e-values of A are all real.

Question 31

When are all the e-values of A imaginary?

Answer 31

When A is a real skew-symmetric matrix.

Question 32

Trick for solving polynomials of degree >= 3.

Answer 32

The only integer roots of the polynomial are factors of its constant term.

Question 33

T or F. If we take the e-values of A, and find the non-zero e-vectors corresponding to these eigenvalues. These e-vectors are linearly dependent.

Answer 33

False.

Question 34

T or F. If we take the e-values of A (and they are distinct), and find the non-zero e-vectors corresponding to these eigenvalues. These e-vectors are linearly dependent.

Answer 34

True.

Question 35

Define diagonalizable.

Answer 35

Let A ∈ Mn(F), there exists B ∈ Mn(F) s.t. B^-1AB = diagonal matrix in Mn(F).

Question 36

T or F. Let A ∈ Mn(F), A can be diagonalizable even if the e-vectors of A do not form a basis of F^n.

Answer 36

False.

Question 37

T or F. If A has n distinct e-values that are all in F, then A is diagonalizable.

Answer 37

True.

Question 38

T or F. If A is diagonalizable, then A must have distinct e-values.

Answer 38

False.

Question 39

Define eigenbasis of F^n for A.

Answer 39

A basis of F^n consisting of e-vectors of a matrix A ∈ Mn(F)

Question 40

Let (v1,...,vn) be an e-basis of F^n for A. Let B = [v1,...,vn]. Then B is invertible and B^-1AB = what? What is the matrix B?

Answer 40

diagonal matrix whose diagonal entries are the e-values of A. The CBM, from basis (e1,...,en) of F^n to the e-basis (v1,...,vn) of F^n for A.

Question 41

What can you say about (B^-1AB)^m?

Answer 41

= B^-1A^mB

Question 42

Let A,C ∈ Mn(F) with equal characteristic polynomials, and Pa(x) has n different roots, what can we say?

Answer 42

Let D be the diagonal matrix with entries of e-values. there exists B s.t. B^-1AB = D and B^-1CB = D Then there exists E s.t. E^-1AE = C. So A ~ C.

Question 43

Let A ∈ Mn(F) with e-values λ1,...,λn all in F. Let r be a positive integer, what are e-values of A^r?

Answer 43

λ1^r,...,λn^r

Question 44

Let A ∈ Mn(F) with distinct e-values λ1,...,λr of A assumed to be in F. Let B1, ... , Br be bases of the e-spaces Esubλ1,...,Esubλr. What can we say about these bases?

Answer 44

If i != j Bi ∩ Bj = ∅ and B1 U ... U Br is a linearly independent subset of F^n.

Question 45

What are two conditions for A to be diagonalizable? Let A ∈ Mn(F).

Answer 45

(a) the sum of the algebraic multiplicities of all e-values of A in F is equal to n. (b) the geometric multiplicity of each e-value equals its algebraic multiplicity.

Question 46

Summarize diagonalization algorithm.

Answer 46

Compute e-values. 1. They must all be in F or we stop. 2. Compute e-space and a basis for each. If all e-values are distinct continue. If all e-values not distinct but the following hold continue: (a) the sum of the algebraic multiplicities of all e-values of A in F is equal to n. (b) the geometric multiplicity of each e-value equals its algebraic multiplicity. Any other case stop. 3. The set B = Bλ1 U ... U Bλr is an e-basis of F^n for A. Let B ∈ Mn(F) be the matrix whose columns are the vectors in B such that the first gλ1 columns are the vectors in Bλ1, then B^-1AB = diag(λ1,...λ1,λ2,...,λ2,...,λr,...,λr). where each n 1<=n<=r has gλ1 copies, gλ2 copies gλr copies

Question 47

List effect on determinant for different EROs.

Answer 47

Adding scalar multiple of another row = *1 Row Switch = *-1 Scalar multiplication = *scalar

Question 48

Let B = kA. We have det(A), how can we get det(B)?

Answer 48

det(B) = k^n * det(A)

Question 49

If |σ(A)| = 1 then? If |σ(A)| = n then? For each state e-values, algebraic multiplicity and characteristic polynomial.

Answer 49

A has just one e-value λ and aλ = n. Pa(x) = (x - λ)^n. A has n distinct e-values λ1,...,λn and aλ1 = ... = aλn = 1. Pa(x) = (x - λ1)(x - λ2)...(x - λn)