Other vocab

Question 1

eigenvalues and eigenvectors of A

Answer 1

For an n × n matrix A, scalars λ and vectors xn×1 ̸= 0 satisfying Ax = λx

Question 2

eigenpair of A

Answer 2

Any such pair, (λ,x), of eigenvalues

Question 3

eigenspace of A

Answer 3

N(A-λI)

Question 4

left hand eigenvectors

Answer 4

Nonzero row vectors y∗ such that y∗(A − λI) = 0

Question 5

characteristic polynomial of A(nxn)

Answer 5

p(λ) = det (A − λI). The degree of p(λ) is n, and the leading term in p(λ) is (−1)^nλ^n.

Question 6

characteristic equation of A

Answer 6

p(λ) = 0

Question 7

principal submatrix

Answer 7

an r × r principal submatrix of An×n is a submatrix that lies on the same set of r rows and columns

Question 8

principal minor

Answer 8

an r × r principal minor is the determinant of an r × r principal submatrix.

Question 9

symmetric function

Answer 9

The kth symmetric function of λ1, λ2, . . . , λn is defined to be the sum of the product of the eigenvalues taken k at a time. That is,sk = 􏰖the sum of λi1 ···λik.

Question 10

spectral radius

Answer 10

For square matrices A, the number ρ(A) = max |λ|

Question 11

Gerschgorin Circles

Answer 11

|z-a(ii)| < or equal to r(i), where r(i)=the sum of |a(ij)| for j=1..n

Question 12

spectrum of A

Answer 12

the set of distinct eigenvalues, denoted by σ (A)

Question 13

algebraic multiplicity

Answer 13

the number of times repeated

Question 14

diagonally dominant

Answer 14

|a(ij)| > the sum of |a(ij)| for each i=1..n, Gerschgorin’s theorem guarantees that diagonally dominant matrices
cannot possess a zero eigenvalue. But 0 ∈/ σ (A) if and only if A is nonsingular, so Gerschgorin’s theorem provides an alternative to the argument used to prove that all diagonally dominant matrices are nonsingular.

Question 15

Similar

Answer 15

Two n × n matrices A and B are said to be similar whenever there exists a nonsingular matrix P such that P^−1AP = B.

Question 16

Similarity transformation on A

Answer 16

The product P^−1AP

Question 17

diagonalizable

Answer 17

A square matrix A is said to be diagonalizable whenever A is similar to a diagonal matrix.

Question 18

complete set of eigenvectors for An×n

Answer 18

any set of n linearly independent eigenvectors for A. Not all matrices have complete sets of eigenvectors

Question 19

deficient

Answer 19

Matrices that fail to possess complete sets of eigenvectors

Question 20

algebraic multiplicity of λ

Answer 20

the number of times it is repeated as a root of the characteristic polynomial. In other words, algmultA(λi) = ai if and only if (x−λ1)^a1 ···(x−λs)^as =0 is the characteristic equation for A.

Question 21

simple eigenvalue

Answer 21

When alg multA (λ) = 1

Question 22

geometric multiplicity of λ

Answer 22

is dim N (A − λI). In other words, geo multA (λ) is the maximal number of linearly independent eigenvectors associated with λ.

Question 23

semisimple eigenvalues of A

Answer 23

Eigenvalues such that alg multA (λ) = geo multA (λ). a simple eigenvalue is always semisimple, but not conversely.

Question 24

spectral decomposition of A

Answer 24

The expansion of the matrix

Question 25

spectral projectors associated with A

Answer 25

the Gi ’s

Question 26

Differential Equations

Answer 26

If An×n is diagonalizable with σ (A) = {λ1, λ2, . . . , λk} , then the unique solution of u′ = Au, u(0) = c, is given by
u=e^(At)c=e(λ1t)v1 +e(λ2t)v2 +···+e(λkt)vk
in which vi is the eigenvector vi = Gic, where Gi is the ith spectral projector.

Question 27

stable system

Answer 27

If Re(λi) <0 foreach i, then lim e^At =0, and limu(t)=0 
for every initial vector c. In this case u′ = Au is said to be a stable system, and A is called a stable matrix.

Question 28

unstable

Answer 28

If Re (λi) > 0 for some i, then components of u(t) can become unbounded as t → ∞, in which case the system u′ = Au as well as the underlying matrix A

Question 29

semistable

Answer 29

If Re (λi) ≤ 0 for each i, then the components of u(t) remain finite for all t, but some can oscillate indefinitely.

Question 30

normal matrix

Answer 30

A ∈ Cn×n is unitarily similar to a diagonal matrix (i.e., A has a complete orthonormal set of eigenvectors) if and only if A∗A = AA∗, in which case A is said to be a normal matrix.