Eigenvectors and Eigenvalues Flashcards Preview

MATH2080 Further Linear Algebra > Eigenvectors and Eigenvalues > Flashcards

Flashcards in Eigenvectors and Eigenvalues Deck (23):
1

Eigenvalue
Definition

-a scalar λ∈F is an eigenvalue for T is there exists a non-zero vector v∈V such that:
T(v) = λv

2

Eigenvector
Definition

-a vector v∈V is an Eigen vector for T if for some λ∈F we have:
T(v) = λv

3

Eigenspace
Definition

-if λ∈F the λ-eigenspace of T is;
Vλ = {v | T(v) = λv}
= {v∈V | T(v) = λI(v)}
= {v∈V | (T-λI)v = 0 }
= ker (T - λI)
-hence Vλ is a subspace of V (since it is the kernel of a linear map)

4

Eigenvalue and Eigenspace Lemma

λ is an eigenvalue of T:V->V
<=>
Vλ ≠ {|0}

5

Eigenvector
Matrix Definition

-let A be an nxn matrix
-a column vector v∈F^n is called an eigenvector of A if for some λ∈F :
Av = λv , i.e. if (A-λI)v=0

6

Eigenvalue
Matrix Definition

λ∈F is an eigenvalue of A if a non-zero column vector v∈F^n exists with:
Av = λv

7

Eigenspace
Matrix Definition

-if λ∈F, the eigenspace Vλ of λ is the null space of A-λI i.e. :
Vλ = {v∈F^n | Av = λv } = {v∈F^n | (A-λI)v = 0}

8

Characteristic Polynomial
Definition

-the characteristic polynomial of an nxn matrix A is;
X(t) = det (A-tI)
= (-1)^n t^n + Cn-1*t^(n-1) + ... + C1*t + Co
-X(t) is a polynomial of degree n
-an equivalent convention is X(t) = det (tI-A), they only differ by a factor of -1 so have the same roots

9

How to find the characteristic polynomial of a matrix?

1) recall, X(t) = det (A-tI) and sub in
2) find the determinant
3) simplify to a polynomial of degree n (where A is an nxn matrix)

10

Characteristic Polynomial and Eigenvalues Theorem

-let A be a square matrix with entries in F
-the eigenvalues of F are the roots of its characteristic polynomial i.e.
λ is an eigenvalue of A <=> Xa(λ)=0
Proof:
λ is an eigenvalue of A <=> Av = λv for some non-zero v
<=> (A-λI)v=0
<=> the matrix A-λI is not invertible (i.e. is singular)
<=> det(A-λI) = 0
<=> Xa(λ)=0

11

Geometric Multiplicity
Definition

-let λ be an Eigen value of nxn matrix A with entries in F, then there exists a column vector v∈F^n such that Av = λv , so (A-λI)v=0
-the geometric multiplicity of λ is the dimension of the λ-eigenspace of A = dim {v∈F^n | (A-λI)v=0}

12

Algebraic Multiplicity
Definition

-let λ be an Eigen value of nxn matrix A with entries in F, then there exists a column vector v∈F^n such that Av = λv , so (A-λI)v=0
-the algebraic multiplicity of λ is the multiplicity of λ as a root of the characteristic polynomial Xa(t), i.e. the number of times it is repeated as a root of the characteristic polynomial

13

How to find the eigenvalues and eigenvectors of a matrix A?

1) find the characteristic polynomial using the equation det(A-λI) = 0
2) the roots of this equation are the eigenvalues
3) one at a time substitute each eigenvalue into the equation (A-λI)v=0 to find the corresponding eigenvector for each eigenvalue

14

How to find algebraic and geometric multiplicity?

1) find the eigenvalues and eigenvectors
2) the number of time each eigenvalue is repeated as a root of the characteristic polynomial is its geometric multiplicity
3) write the eigenvector of each eigenvalue as a span and then find the dimension, this is the algebraic multiplicity

15

Diagonalisable
Definition

-a matrix A is diagonalisable if it is similar to a diagonal matrix i.e. A is diagonalisable if there exists a non-singular (invertible) matrix P such that P^(-1) A P = Λ with Λ diagonal
-where the columns of P correspond to the eigenvectors of the eigenvalues that form the diagonal of Λ, i.e. the nth column of P is the eigenvector that corresponds to the eigenvalue in the nth column of Λ

16

Diagonalisability Equivalence Theorem

-let A be an nxn matrix, the following are equivalent:
i) A is diagonalisable
ii) A has n linearly independent eigenvectors
- an additional statement of equivalence can be made if A is a matrix over the field of complex numbers, C :
iii) for all eigenvalues, the geometric multiplicity is equal to the geometric multplicity

17

How to diagonalise a matrix?

1) find n linearly independent eigenvectors (if this is not possible then the matrix is not diagonalisable)
2) columns of P = the eigenvectors of A
3) Λ = nxn diagonal matrix with entries corresponding to the eigenvalues of the eigenvectors in P
4) P^(-1) A P = Λ

18

Eigenspaces Lemma

-let A be an nxn matrix with entries in F, λ1,...,λm are the distinct eigenvalues of A
-for i = 1 to m consider basis Ri of the eigenspace:
Vλi = { v∈F^n | (A - λi I ) |v = |0 }
-the set R1∩R2∩R3∩...∩Rm obtained by putting together all of the bases Ri of eigenspaces Vλi is linearly independent

19

Eigenspaces Lemma
Special Case n distinct eigenvalues

-let A be an nxn matrix with entries in F, λ1,...,λm are the distinct eigenvalues of A
-non-zero eigenvectors v1,v2,...,vm associated to different eigenvalues λ1,...,λm of a matrix are linearly independent
-thus if nxn matrix A has n distinct eigenvalues then A is diagonalisable since it will also have n linearly independent eigenvectors

20

Properties of Matrix Determinants

-if A, B are nxn matrices, then det(AB) = detA detB
-if A is invertible (detA ≠ 0) then det(A^(-1)) = 1/ detA
-similar matrices A and B have the same determinant since B = PAP^(-1)
det B = det (PAP^(-1)) = detP det A det (P^(-1)) = detPdetA/detP = detA

21

Similar Matrices Characteristic Polynomial Theorem

-similar matrices (A & B) have the same characteristic polynomial:
Χb(t) = det(B - tI) = det(P^(-1)AP - tP^(-1)P) =
det(P^(-1)(A-t)P) = det(P^(-1)) det(A-It) det(P) = det(A-tI) = Xa(t)
-hence if Χb(t) ≠ Xa(t) , then matrices A and B cannot be similar

22

Cayley-Hamilton Theorem
2x2 Matrices

-let A be a 2x2 matrix
-let p(t) = Xa(t) = t² + mt + n be the characteristic polynomial of A
-then A is a root of it characteristic polynomial:
p(A) = A² + mA + nI = |0
-where I is the 2x2 identity matrix

23

Cayley Hamilton Theorem
nxn Matrices

-let A be an nxn matrix
-let p(t) = t^n + a_n-1 * t^(n-1) + ... + a2*t² + a1*t + ao be the characteristic polynomial of A
-then p(A) = 0 where 0 is the nxn zero matrix:
A^n + a_n-1*A^(n-1) + ... + a2*A² + a1*A + a0*I = |0
-where I is the nxn identity matrix