Eigenvectors / Eigen Values (Cayley Hamilton)

Question 1

eigenvector/eigenvalue of a matrix

Answer 1

Is A is an nxn matrix over a field 𝔽 and v∈𝔽ⁿcol is non zero then v is an eigenvector with eigenvalue λ when
Av=λv
(where A is a matrix that represents a linear map)

Question 2

An eigenvector has…

Answer 2

exactly one eigenvalue

Question 3

An eigenvalue has…

Answer 3

many possible eigenvectors

Question 4

if v is an eigenvector then…

Answer 4

every multiple of v is also an eigenvector

Question 5

0 can and cant be…

Answer 5

0 can be an eigenvalue

0 can’t be an eigenvector

Question 6

eigenspace

Answer 6

If λ is an eigenvalue, the corresponding eigenspace is given by key(f-λI) and is the set of all non zero eigenvectors corresponding to λ.

Question 7

Eigenvalues are the roots of the equation…

Answer 7

det(f-xI) = 0

Question 8

each eigenspace has dimension…

Answer 8

atleast 1

Question 9

0 is an eigenvalue of f iff…

Answer 9

iff the nullity of f is non-zero

Question 10

Lemma 9.3. eigenvalue of a square matrix.

Answer 10

A scalar λ is an eigenvalue of a square matrix A if and only if n(A-λI)>0

Question 11

Corollary 9.5

Answer 11

A scalar λ∈ℂ is an eigenvalue of an nxn square matrix A (over ℂ or ℝ) iff det(A-λI)=0

Question 12

If det(A-λI)=0 then…

Answer 12

there are infinitely many eigenvectors for λ

Question 13

How to find eigenvalues

Answer 13

solve det(A-xI)=0

Question 14

How to find eigenvectors

Answer 14

1. find A-λI
2. put into echelon form
3. find ker(A-λI). Extract equations from this form.
4. if equations each in terms of two variables only, put into a vector
5. otherwise let x = equation in terms of other two variables. let a=y, b=z and write a () + b () where you put 1s in correct position for y and z in each and in x position put the coefficients of the equation.

Question 15

if λ is an eigenvalue of a then:

Answer 15

1. there is a v s.t. Av = λv
2. dim ker (A - λI)>0
3. nullity (A- λI) > 0
4. rank (A - λI) < n
5. det(A-λI)=0 (i.e. matrix is not invertible)

Question 16

Characteristic polynomial of A

Answer 16

χₐ(x) = det(A - xI)

Question 17

Proposition 10.1. eigenvalues in UT matrices.

Answer 17

The eigenvalues are the diagonal entries of fᵢᵢ.

Question 18

The characteristic polynomial can also be written as…

Answer 18

χₐ(x) = Π(fᵢᵢ-x)

Question 19

Powers of A…

Answer 19

commute

Question 20

Polynomials in A….

Answer 20

commute

Question 21

Theorem 10.3. Lemma for cayley Hamilton.

Answer 21

If A is upper triangular then the characteristic polynomial evaluated at A is 0.

Question 22

proposition 10.5. similar matrices and polynomials.

Answer 22

if p(x) is a polynomial over 𝔽 and A,B are similar then p(A) and P(B) are similar

Question 23

Proposition 10.6. similar matrices are their characteristic polynomials.

Answer 23

if B = P⁻¹AP then χₐ(x) = χb(x)

Question 24

Theorem 10.7 (beore cayley hamilton)

Answer 24

Let A be a nxn square matrix over ℂ then A is similar to an upper triangular matrix.

Question 25

Theorem 10.10 (the cayley hamilton theorem)

Answer 25

Let A be an nxn square matrix over ℂ or ℝ. Then χₐ(A)= det(A-A) = det(0) = 0

Question 26

Theorem 10.11 (after cayley hamilton)

Answer 26

if χₐ(x) = (λ-x)ᵉp(x) where λ is not a root of p(x) then the nullity of the matrix (A-λI)ᵉ is e

Question 27

How to find the determinant of a 2x2 matrix (a b , c d)

Answer 27

ad-bc

Question 28

How to find the determinant of a 3x3 matrix (a b c , d e f , g h i)

Answer 28

expand which whichever column/row you want to e.g.

a | e f , h i | + b | f d , i g | + c | d e , g h |

Question 29

how to find the inverse of a 3x3 matrix (a b c , d e f , g h i)

Answer 29

1. find the determinant
2. find all the cofactors of the matrix
3. mirror across the diagonal
4. divide by the determinant

Question 30

how to find the inverse of a 2x2 matrix (a b , c d)

Answer 30

1. find the determinant
2. mirror the diagonal and switch non diagonal elements signs
3. divide by the determinant

Question 31

How to show a matrix can’t be put into upper triangular form (show A is not similar to an upper triangular matrix)

Answer 31

1. find P, and its inverse
2. calculate P⁻¹AP (or just calculate the bottom left element of P⁻¹AP)
3. either its non zero so not upper triangular or, if you don’t know P show that the equation can’t be solved.

Question 32

How to find the upper triangular form for a matrix

Answer 32

1. find eigenvalues of A
2. find the eigenspaces of A
3. if the multiplicity of each element of χₐ(x) is not equal to the corresponding dimension of the eigenspace then can’t diagonalise. Find the eigenspaces for each eigenvalue.
4. for the eigenvalue where multiplicty isnt equal to dimenision look at im(A-λI) (column space)
5. find 3 spanning vectors for A (apply column operations to find possible vectors)
6. Find A w.r.t these bases and write as a linear combination then put into the the matrix.