Other vocab Flashcards
eigenvalues and eigenvectors of A
For an n × n matrix A, scalars λ and vectors xn×1 ̸= 0 satisfying Ax = λx
eigenpair of A
Any such pair, (λ,x), of eigenvalues
eigenspace of A
N(A-λI)
left hand eigenvectors
Nonzero row vectors y∗ such that y∗(A − λI) = 0
characteristic polynomial of A(nxn)
p(λ) = det (A − λI). The degree of p(λ) is n, and the leading term in p(λ) is (−1)^nλ^n.
characteristic equation of A
p(λ) = 0
principal submatrix
an r × r principal submatrix of An×n is a submatrix that lies on the same set of r rows and columns
principal minor
an r × r principal minor is the determinant of an r × r principal submatrix.
symmetric function
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.
spectral radius
For square matrices A, the number ρ(A) = max |λ|
Gerschgorin Circles
|z-a(ii)| < or equal to r(i), where r(i)=the sum of |a(ij)| for j=1..n
spectrum of A
the set of distinct eigenvalues, denoted by σ (A)
algebraic multiplicity
the number of times repeated
diagonally dominant
|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.
Similar
Two n × n matrices A and B are said to be similar whenever there exists a nonsingular matrix P such that P^−1AP = B.
Similarity transformation on A
The product P^−1AP
diagonalizable
A square matrix A is said to be diagonalizable whenever A is similar to a diagonal matrix.
complete set of eigenvectors for An×n
any set of n linearly independent eigenvectors for A. Not all matrices have complete sets of eigenvectors
deficient
Matrices that fail to possess complete sets of eigenvectors
algebraic multiplicity of λ
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.
simple eigenvalue
When alg multA (λ) = 1
geometric multiplicity of λ
is dim N (A − λI). In other words, geo multA (λ) is the maximal number of linearly independent eigenvectors associated with λ.
semisimple eigenvalues of A
Eigenvalues such that alg multA (λ) = geo multA (λ). a simple eigenvalue is always semisimple, but not conversely.
spectral decomposition of A
The expansion of the matrix
spectral projectors associated with A
the Gi ’s
Differential Equations
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.
stable system
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.
unstable
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
semistable
If Re (λi) ≤ 0 for each i, then the components of u(t) remain finite for all t, but some can oscillate indefinitely.
normal matrix
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.
