eigensystems and canonical forms take 2

Question 1

eigenvector:

Answer 1

a vector x is an eigenvector of A (a square matrix) if x is nonzero and Ax=λx where λ is a scalar

Question 2

eigenvalue:

Answer 2

a scalar λ is an eigenvalue of A (a square matrix) if λ is nonzero and Ax=λx where x is nonzero

Question 3

eigenpair:

Answer 3

an eigenvector and its associated eigenvalue, each eigenvector has one eigenvalue but each eigenvalue has many associated eigenvectors

Question 4

characteristic polynomial:

Answer 4

p(λ)=det(λI-A)=0

Question 5

spectrum:

Answer 5

set of all eigenvalues of A, written Λ(A)

Question 6

algebraic multiplicity:

Answer 6

of λ, its multiplicity as a 0 of the characteristic polynomial, which I Think means the number of times λ appears as a root of p(λ)?

Question 7

invariant subspace:

Answer 7

a subspace X in C^n is invariant for A if AX in X, that is, x in X implies Ax in X

Question 8

matrices and subspaces:

Answer 8

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

Question 9

similar:

Answer 9

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

Question 10

similarity transformation:

Answer 10

P^(-1)AP

Question 11

transforming matrix:

Answer 11

P in a similarity transformation

Question 12

unitarily similar:

Answer 12

B=U*AU where U is a unitary matrix

Question 13

orthogonally similar:

Answer 13

A and B are real, B=U^(T)AU where U is a real orthogonal matrix

Question 14

diagonalisable:

Answer 14

if a matrix A is similar to a diagonal matrix, A is diagonalisable or simple

Question 15

schur’s theorem:

Answer 15

let A be square, 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*, equivalent to T=U^(-1)AU

Question 17

schur vectors:

Answer 17

the columns of U

Question 18

normal:

Answer 18

a matrix is normal if AA=AA

Question 19

spectral theorem:

Answer 19

let A be square, then A is normal iff there exists a unitary matrix U and diagonal matrix Λ such that A=UΛU*

Question 20

normalcy and orthogonal vectors:

Answer 20

an nxn matrix A is normal iff it has n orthogonal eigenvectors

Question 21

diagonalisability and eigenvectors:

Answer 21

a square matrix A is diagonalisable iff A has n linearly independent eigenvectors

Question 22

diagonalisability and eigenvalues:

Answer 22

a matrix with distinct eigenvalues is diagonalisable

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 jcl

Question 25

number of jordan blocks:

Answer 25

the number p of jordan blocks is the number of linearly independent eigenvectors of A, so it’s diagonalisable iff p=n

Question 26

algebraic multiplicity of an eigenvalue:

Answer 26

the algebraic multiplicity of a given eigenvalue λ is the sum of dimensions of the jordan blocks in which it appears

Question 27

geometric multiplicity of an eigenvalue:

Answer 27

the geometric multiplicity of a given eigenvalue λ is the number of associated jordan blocks, so the number of associated linearly independent eigenvectors, so dim(null(A-λI))

Question 28

defective (eigenvalue):

Answer 28

if the eigenvalue appears in a jordan block of size greater than 1, or equivalently if its algebraic multiplicity > its geometric multiplicity

Question 29

defective (matrix):

Answer 29

a matrix is defective if it has a defective eigenvalue, or equivalently if it does not have a complete set of linearly independent eigenvectors

Question 30

how to find the jcl (but not the similarity transformation matrices just the jcl):

Answer 30

find all the distinct eigenvalues by finding the roots of the characteristic polynomial or whatever
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 jordan block corresponding to λi, called the index, and the number of jordan 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 31

generalised eigenvectors:

Answer 31

take the X in X^(-1)AX=J (J=the jcl) - the columns of X in positions 1, m1+1, m1+m2+1, …, m1+…+m(p-1)+1 and these are linearly independent eigenvectors of A, the Other columns are the generalised eigenvectors

Question 32

jordan chain:

Answer 32

equating the first m1 columns of X to the first jordan block J1 gives Ax1=λ1x1, Axi=λ1xi+x(i-1), i=2,…,m1 - the vectors x1,x2,…,xm1 are called a jordan chain, the columns of X form p jordan chains - next one is x(m1+1),…,x(m1+m2) and so on

Question 33

cayley-hamilton theorem:

Answer 33

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

Question 34

minimal polynomial:

Answer 34

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