7) Determinants, eigenvalues and eigenvectors

Question 1

What is

Answer 1

The (n − 1) × (n − 1) matrix obtained from M by deleting the i-th row and j-th column

Question 2

What is the determinant of a matrix

Answer 2

Question 3

What is Laplace Expansion Theorem

Answer 3

Question 4

What is the minor and cofactor of a matrix

Answer 4

Question 5

What is an upper triangle matrix

Answer 5

A square matrix M is called upper triangular if Mij = 0 for all i > j,
that is, if all entries below the main diagonal are 0

Question 6

What is the determinant of an upper triangle matrix

Answer 6

Question 7

What is the determinant of an identity matrix

Answer 7

Question 8

Describe how the elementray row operations relate to the determinant

Answer 8

Question 9

When is a matrix invertiable

Answer 9

If det M ≠ 0 M is invertiable

Question 10

Describe the proof that a matrix is invertiable if det M ≠ 0

Answer 10

Question 11

What is the adjugate of a sqaure matrix A

Answer 11

The transpose of the cofactor matrix of A

Question 12

How are matrices and their adjugate related

Answer 12

If A is an invertiable matrix and B is the adujgate of A

Question 13

If A is a matrix and B is the adjugate of A, what is AB

Answer 13

AB = (det A)In = BA

Question 14

What is det(AB)

Answer 14

det(A)det(B)

Question 15

Describe the proof that det(AB) = det(A)det(B)
(Don’t need to know)

Answer 15

Question 16

If we use the LU factorization of a permutation P A of A, then what is det(P)det(A)

Answer 16

• det P det A = det(PA) = det(LU) = det L det U
• Where P = the row interchanges, L = the lower triangular with ones on the diagonal, and U = an upper triangular matrix

Question 17

What is an eigenvector

Answer 17

The vector v ∈ V is an eigenvector of T with eigenvalue
λ ∈ K if v is non-zero and T(v) = λv

Question 18

What is an Eigenspace

Answer 18

E*λ * = {v ∈ V |T(v) = λv}.

Question 19

How are Eigenspaces written

Answer 19

Question 20

What is the geometric multiplicity of an eigenvalue

Answer 20

The dimension of the corresponding λ-eigenspace

Question 21

What is the Linear Independence of Eigenvectors

Answer 21

• Let T : V → V be a linear map. Suppose that v1, . . . , vk are eigenvectors of T with distinct eigenvalues λ1, . . . , λk, respectively
• Then, the set {v1, . . . vk} is linearly independent

Question 22

When is a field algebraically closed

Answer 22

Question 23

Are the set of real numbers algebraically closed

Answer 23

No since the polynomial x^2+1 has no real roots

Question 24

Are the set of complex numbers algebraically closed

Answer 24

Yes

Question 25

What is the algebraic multiplicity of λ (an eigenvalue
of A)

Answer 25

Then the power k to which the factor x − λ appears in the characteristic polynomial pA(x)

Question 26

What is the trace of a matrix

Answer 26

Question 27

If K is an algebraically closed field and λ1, . . . , λn be all the eigenvalues of A ∈ Mn(K), what is the trace and determinant of A

Answer 27

Question 28

Describe the proof that

Answer 28

Question 29

Describe the proof that

Answer 29