Theorems basic+eigen

Question 1

For A ∈ C^n×n

the following conditions are equivalent to A being nonsingular:

Answer 1

1. null(A) = {0} (i.e, there is no nonzero y ∈ C^n such that Ay = 0 ).
2. rank(A) = n (i.e., the rows or columns of A are linearly independent).
3. det(A) not equal to 0.
4. None of A’s eigenvalues is zero.

Question 2

Let the columns of X ∈ C^n×p, p ≤ n, form a basis for a subspace X! of C^n
Then X! is an invariant subspace for A if and only if

Answer 2

AX = XB for some B ∈ C^p×p

When the latter equation holds, the spectrum of B is contained within that of A

Question 3

Let A and B be similar, say B = P−1AP.

Then A and B have the same eigenvalues, and x is an eigenvector of A with associated eigenvalue λ if and only if

Answer 3

P−1x is an eigenvector of B with associated eigenvalue λ.

Question 4

Schur Theorem

Answer 4

Let A ∈ C^n×n
There exists a unitary matrix U and
an upper triangular matrix T such that
T = U−1AU = U*AU

Question 5

Spectral theorem

Answer 5

Let A ∈ C^n×n
A is normal if and only if there is:
Unitary matrix U and a diagonal matrix Λ such that
A = UΛU*

Question 6

A ∈ C^n×n is normal if and only if it has

Answer 6

n orthogonal eigenvectors

Question 7

Let A be an n×n matrix

A is diagonalizable if and only if

Answer 7

A has n linearly independent eigenvectors.

Question 8

A matrix with distinct eigenvalues is

Answer 8

diagonalizable

Question 9

Jordan canonical form

Answer 9

Any matrix A ∈ C^n×n can be expressed in the Jordan canonical form: X−1AX=

[J1(λ1)
         J2(λ2)
                    .
                        .
.
                             Jp(λp)]

Jk = Jk(λk) =
[λk     1
        λk
             .
                        . 1
                        λk]      
∈ C^mk×mk

where X is nonsingular and m1 + m2 + · · · + mp = n

Question 10

Cayley–Hamilton theorem

Answer 10

If p is the characteristic polynomial of an n × n matrix A, then p(A) = 0.

Question 11

Minimal polynomial

Answer 11

Let A be an n × n matrix with s distinct eigenvalues λ1, . . . , λs. The minimal polynomial of A is
q(λ) = PROD^s_i=1 : (λ − λi)^ni

where ni is the dimension of the largest Jordan block in which λi appears