MATH 574 Lecture 3-6 notes.

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*Definition*:
$U \in U_n$ is **unitary** (酉矩阵) if $U^* U = I$.
If U is real, usually say U **orthogonal**.

*Definition*:
Vectors ${x_1, \dots, x_n}$ are an **orthogonal set** if $x_i^* x_j = 0 \forall i \ne j$.
If $||x_i|| = 1$ for any i, then we call it **orthonormal** (标准正交).

*Property*:
An orthogonal set of nonzero vectors is linearly independent.

*Theorem*: TFAE:

- U is unitary;
- U nonsingular and inv(U) = U*;
- UU* = I;
- U* is unitary;
- columns of U orthonormal;
- rows of U* orthonormal;
- u is an isometry: (Ux)* (Ux) = x* x

**Schur’s Lemma**:
Given $A \in M_n$, with eigenvalues $\lambda_1, \dots, \lambda_n$ in any order, then exist unitary U s.t. U*AU = T, with T upper-diagonal with diagonal elements the eigenvalues.
If A real and all its eigenvalues real, then U can be real and orthogonal.

*Theorem*:
Let F be a commuting family, then exist unitary U s.t. U*AU triangular for any A in F.
If F is real with real eigenvalues, we can do it over R, with diagonal blocks 1-by-1 or 2-by-2.

*Property*:
If A has eigenvalues $\lambda_1, \dots, \lambda_n$, B has eigenvalues $\mu_1, \dots, \mu_n$ (including multiplicities), and A, B commute,
then A+B has eigenvalues $\lambda_1 + \mu_\omega(1), \dots, \lambda_n + \mu_\omega(n)$, where $\omega in S_n$.

*Theorem* (**McCoy**): (without proof)
Let $A, B \in M_n$ with eigenvalues $\lambda_1, \dots, \lambda_n; \mu_1, \dots, \mu_n$ respectively (including multiplicities).
Then exist S in **GLn(C)** s.t. both $S^-1 A S$ and $S^-1 B S$ upper triangular iff $\sigma (f(A,B)) = {f(\lambda_i, \mu_\omega(i)) | i=1, \dots, n}$ for some $\omega \in S_n$ and all polynomials $f \in C$.

*Theorem*:
Suppose A has eigenvalues $\lambda_1, \dots, \lambda_k$ occurring $n_1, \dots, n_k$ times.
Then A is similar to $\text{diag}{T_1, T_2, \dots, T_n}$, where $T_i$ is $n_i * n_i$ upper triangular matrix, with diagonal elements $\lambda_i$.
If A real and all its eigenvalues real, this can be done with a real similarity matrix.

*Definition*:
A matrix N is **nilpotent** (幂零矩阵) if $N^k = 0$ for some k.
(If $N \in M_n$ is nilpotent, then $N^n = 0$.)

*Theorem*:
Let A be a strictly upper triangular matrix, then exist S in GLn and $n_1 \ge \dots \ge n_m > 0$ with $\Sigma n_i = n$ s.t. $S^{-1} A S = J_{n_1}(0) \oplus \dots \oplus J_{n_m}(0)$.
Moreover, if A real, then S in GLn(R).

Proof: Induction.

*Theorem*: (**Jordan Form**)
Let $A \in M_n$, then exist S in GLn(C), s.t. $S^-1 A S = J_{n_1}(\lambda_1) \oplus \dots \oplus J_{n_k}(\lambda_k)$.
If A real and all its eigenvalues real, then S can be real.

Proof: Existence is now obvious. For uniqueness, consider $\text{rank}(J- \lambda I)^m$.

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*Definition*:
A matrix is **nonderogatory** if every eigenvalue has geometric multiplicity 1.

*Theorem*:
A is nonderogatory iff that commutes with A is a polynomial in A.
(Claim: If S^-1 B S = f(J), then B = f(A).)
(The inverse is not proved on textbook.)

*Theorem*:
Every matrix A can be written $A = A_D + A_N$, where $A_D$ diagonalizable, $A_N$ nilpotent, and $A_D$ and $A_N$ commute.

*Corollary*:
$A^m \to 0$, as $m \to \infty$ iff $|\lambda| <1$ for any eigenvalue of A.

*Definition*: An **annihilatory polynomial** is any $g \in C[t]$ s.t. $g(A) \equiv 0$.

*Definition*:
The **minimal polynomial** of A, $m_A(t)$, is the unique monic (首一) polynomial of minimum positive degree that annihilates A.

*Theorem*: $m_A$ is well defined.

(Proof: polynomial division)

*Corollary*:
The minimal polynomial divides the characteristic polynomial and has all the same roots.

*Property*: TFAE:

- A diagonalizable
- m_A splits into distinct linear factors
- every root of m_A has multiplicity 1
- any t_0 s.t. m_A(t_0) =0, m^1_A(t_0) != 0

*Definition*:
Let $f(t) = t^n + a_{n-1} t^{n-1} + \dots + a_0$, **companion matrix** of f is
$\begin{bmatrix}
0 & - a_0 \\
I & - a_{n-1}
\end{bmatrix}$.

*Property*:
Every monic polynomial is the minimal and characteristic polynomial of its companion.

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P = {Polynomials in one variable t over C}. This is a vector space.

*Definition*:
Span{basis} == {finite linear combinations of {basis}}

Note: $(1, \dots) \notin \text{span}{e_i}$

*Definition*:
U = {all unitary matrices}.

(Note: U is a compact subgroup of GLn(C).)

Table: Comparison of concepts in Real and Complex field

C | R |
---|---|

Unitary | Orthogonal |

Hermitian | Symmetric |

skew Hermitian | skew Symmetric |

*Property*:
Let H be Hermitian:

- $\forall x \in C^n, x^* H x \in R$;
- eigenvalues of H are real;
- eigenvectors associated with different eigenvalues are orthogonal. (By orthogonal in C, we mean $x^* y = 0$)

*Theorem* (**Spectral theorem** / **Principal axis theorem**):
A Hermitian (symmetric) can be diagonalized by unitary (orthogonal) matrix.

(Matrices equivalent to a Hermitian matrix is Hermitian. ?)

Note: A real matrix with real eigenvalues has real eigenvectors, they are not orthogonal unless A is symmetric.

*Property*:
If A Hermitian and $x^* A x \ge 0 \forall x \in C_n$, then all eigenvalues of A are (real) nonnegative.

*Property*:
Every $A \in M_n(C)$ has a unique decomposition as $A = H + iH'$, with H and H' Hermitian.

*Theorem* (proof not shown):
Let F be a family of Hermitian matrices, then exist U unitary s.t. $U^* A U$ diagonal for any A in F iff F is a commuting family.

*Definition*:
A matrix is **normal** (正规) if it commutes with its adjoint (共轭转置).

(Note: Similarity does not guarantee same eigenvectors.)

*Theorem* (**Spectral theorem for normal matrices**):
Let $A in M_n$ with eigenvalues $\lambda_1, \dots, \lambda_n$, TFAE:

- A is normal;
- A is unitarily diagonalizable;
- $\Sigma |a_{ij}|^2 = \Sigma |\lambda_i|^2$
- exist orthonormal set of n eigenvectors of A.

Similarity transformations:

- A arbitrary: $M^-1 A M = J$, with columns of M eigenvectors & "generalized eigenvectors", and J Jordan form.
- A diagonalizable: $S^-1 A S = \Lambda$, with columns of S eigenvectors and $\Lambda$ diagonal.

**Schur’s Lemma**:

- A arbitrary: exist unitary U s.t. $U^* A U = T$, with T upper triangular.
- A normal: exist unitary U s.t. $U^* A U = \Lambda$, with $\Lambda$ diagonal.

*Theorem* (Proof not shown):
Every matrix is similar to a symmetric matrix.
("Symmetric Jordan form")

Special cases of normal matrices:

- Hermitian: $\Lambda$ is real.
- real symmetric: $\Lambda$ is real, and U is (real) orthogonal matrix.
- skew hermitian: $\Lambda$ is pure imagery;
- unitary: norm of eigenvalue = 1;

6 Hermitian matrices

*Definition*:
**Rayleigh-Ritz ratio**: $\frac{x^* A x}{x^* x}$

*Theorem* (**Rayleigh-Ritz**):