eigensystems and canonical forms

Question 1

eigenvector:

Answer 1

a vector x is an eigenvalue of A (A is square) if x is nonzero and Ax is a multiple of x
i.e. there is some λ such that Ax=λx

Question 2

eigenvalue:

Answer 2

λ is an eigenvalue of A if Ax=λx with x being an eigenvector
det(λI-A)=0

Question 3

eigenpair:

Answer 3

the λ and x that are eigenvalue/vector of A

Question 4

characteristic polynomial:

Answer 4

p(λ)=det(λI-A) - some polynomial equal to 0

Question 5

spectrum:

Answer 5

the set of all eigenvalues of A
an nxn matrix has n eigenvalues, so Λ(A)={λ1,…,λn}

Question 6

invariant subspace:

Answer 6

a subspace X is invariant for A if AX in X - x in X implies Ax in X

Question 7

how to check if the spectrum of B is contained within the spectrum of A:

Answer 7

let the columns of Y in C^(nxp), p<=n, form a basis for a subspace X of C^n. X is invariant iff AY=YB for some B in C^pxp. when the latter holds, the spectrum of B is contained within that of A

Question 8

similar:

Answer 8

two matrices A and B are similar if there exists a nonsingular matrix P such that B=P^(-1)AP

Question 9

similarity transformation:

Answer 9

B=P^(-1)AP

Question 10

transforming matrix:

Answer 10

P in B=P^(-1)AP

Question 11

similarity and eigenpairs:

Answer 11

if A and B are similar, A and B have the same eigenvalues, and x is an eigenvector of A with associated eigenvalue λ iff P^(-1)x is an eigenvector of B with the same associated eigenvalue

Question 12

unitarily similar:

Answer 12

two matrices A and B are unitarily similar if there is a unitary matrix U such that B=U*AU

Question 13

orthogonally similar:

Answer 13

if A and B are real matrices, they are orthogonally similar if there is a real orthogonal matrix U such that B=U^(T)AU

Question 14

if A is similar to a diagonal matrix:

Answer 14

A is called diagonalizable and/or simple

Question 15

schur’s theorem:

Answer 15

let A be a square matrix, then there exists a unitary matrix U and an upper triangular matrix T such that T=U^(-1)AU=U*AU

Question 16

schur decomposition:

Answer 16

A=UTU*, not unique

Question 17

schur vector:

Answer 17

the columns of U in a schur decomp

Question 18

normal:

Answer 18

a matrix A is normal if AA=AA

Question 19

the spectral theorem:

Answer 19

god what a cool name
A is normal iff there is a unitary matrix U and a diagonal matrix Λ such that A=UΛU*

Question 20

if A in C^nxn has n orthogonal eigenvectors:

Answer 20

A is normal

Question 21

an nxn matrix A is diagonalizable iff:

Answer 21

A has n linearly independent eigenvectors

Question 22

a matrix with distinct eigenvalues is:

Answer 22

diagonalizable

Question 23

the jordan canonical form:

Answer 23

any square matrix can be expressed in the form X^(-1)AX=J=
[J1(λ1)
…
Jp(λp)]
Jk=Jk(λk)=
[λk 1
λk 1
… …
1
Ak] in C^mkxmk
where X is nonsingular and m1+…+mp=n

Question 24

jordan block:

Answer 24

the mkxmk matrices in the jcf that go along the diagonal

Question 25

jordan matrix properties:

Answer 25

the number p of jordan block is the number of linearly independent eigenvectors of A, so A is diagonalizable iff n=p
the algebraic multiplicity of an eigenvalue λ is the sum of dimensions of the jordan blocks in which λ appears
the geometric multiplicity of λ is the number of jordan blocks associated with λ - dim(null(A-λI))

Question 26

defective:

Answer 26

an eigenvalue is defective if it appears in a jordan block of size greater than 1, or equivalently if its algebraic multiplicity exceeds its geometric multiplicity
a matrix is defective if it has a defective eigenvalue or equivalently if it doesn’t have a complete set of linearly independent eigenvectors

Question 27

how to find the jordan canonical form of a given matrix A:

Answer 27

find all the distinct eigenvalues of A, maybe by finding the roots of the characteristic polynomial
for each distinct eigenvalue λi of A, form (A-λiI), (A-λiI)^2, … and analyse the sequence of ranks as follows:
the smallest value of ki for which rank(A-λiI)^ki attains its minimum value is the order of the largest block corresponding to λi, this is called the index of λi
the no. of blocks of size k in J with eigenvalue λi is rank(A-λiI)^(k-1)+rank(A-λiI)^(k+1)-2rank(A-λiI)^k

Question 28

generalised eigenvectors:

Answer 28

let X^(-1)AX=J
AX=XJ
the columns of X in positions 1, m1+1, m1+m2+1,… are eigenvectors of A and are linearly independent as X is nonsingular
the other columns are generalised eigenvectors

Question 29

jordan chain:

Answer 29

equating the first m1 columns of AX=XJ corresponding to the first jordan block J1 gives us Ax1=λ1x1, Axi=λ1xi+x(i-1), i=2,…,m1
the vectors x1,…,xm1 are called a jordan chain

Question 30

cayley-hamilton theorem:

Answer 30

if p is the characteristic polynomial of a nxn matrix A, then p(A)=0

Question 31

minimal polynomial:

Answer 31

let A be a nxn matrix with s distinct eigenvalues λ1,…,λs - the minimal polynomial is q(λ)=(s)Π(i=1)(λ-λi)^ni, where ni is the dimension of the largest jordan block in which λi appears