Special Matrices 特殊矩阵与算子

正规矩阵、厄米矩阵、酉矩阵、反厄米矩阵等性质

1 Normal Operator

Definition: Normal Operator

Definition 1 \(A \in \mathbb{C}^{n \times n}\) is called normal iff \(A^H A = AA^H\)

1.1 Notes

1. \(\iff\) \(A\) is unitarily-diagonalizable, i.e., \[ \exists U \in U(n), \text{diagonal } D, \text{s.t. } A = UDU^H. \quad (\text{Spectral Thm 2}) \]

   \(\iff\) \(\exists \text{ orthogonal eigenvectors that span } \mathbb{C}^n.\)

   \(\iff\) \(A \text{ orthogonally non-defective}\)

   \(\iff\) \(\text{row covariance matrix of }A = \text{column covariance matrix of }A\)

2. Normal matrix \(A\) does not necessarily invertible. Its eigenvalues can be \(0, \mathbb{R}\) or \(\mathbb{C}\).

3. Mental picture for normal operator:

   \[ \boxed{\text{Normal operators} \iff \text{Squeezing complex rectangular box.}} \]

2 Hermitian (Self-adjoint) Operator

Definition: Hermitian Operator

Definition 2 \(A \in \mathbb{C}^{n \times n}\) is called hermitian (self-adjoint) iff \(A = A^H\)

2.1 Notes

1. \[ A \text{ hermitian} \iff \begin{equation*} \begin{cases} A \text{ normal} & \\ \text{spectrum of }A \subseteq \mathbb{R}. & \end{cases} \end{equation*} \]

2. Also self-adjoint operators do not necessarily invertible.

3. Self-adjoint operators could be think of as normal operators with real spectrum.

4. Mental picture for hermitian operators:

   \[ \boxed{\text{Hermitian operators} \iff \text{Squeezing complex rectangular box in a particular way that creatures living under projection }\pi: \mathbb{C}^n \to \mathbb{R}^n \text{ do NOT think it is a rotation.}} \]

5. If \(A \in \mathbb{R}^{n\times n} < \mathbb{C}^{n\times n}\) is hermitian, it is also called symmetric, i.e., \[ A = A^t. \]

3 Skew-Hermitian Operator

Definition: Hermitian Operator

Definition 3 \(A \in \mathbb{C}^{n \times n}\) is called skew-hermitian iff \(-A = A^H\)

3.1 Notes

1. Skew-hermitian operators are very close to hermitian: \[ A \text{ skew-hermitian} \iff iA \text{ hermitian} \]

Proof

Let \(A \in \mathbb{C}^{n \times n}\) be skew-hermitian, i.e., \(-A = A^H\). Let \(B = iA\). Then \[ B^H = (iA)^H = -iA^H = iA = B. \]

Therefore, \(B\) is hermitian.

2. By the property that hermitian operators have real spectrum, all skew-hermitian operators have purely imaginary spectrum.

3. \[ A \text{ skew-hermitian} \iff \begin{equation*} \begin{cases} A \text{ normal} & \\ \text{spectrum of }A \subseteq i\mathbb{R}. & \end{cases} \end{equation*} \]

4 Unitary Operator

Definition: Unitary Operator

Definition 4 \(A \in \mathbb{C}^{n \times n}\) is called unitary iff \(AA^H = I\), denoted \(A \in U(n)\)

Proposition

Proposition 1 \[\begin{equation*} \begin{cases} A \text{ square} & \\ A^HA = I & \end{cases} \implies A A^H = I \end{equation*}\]

Proof. From \(A^H A = I\) we know that the columns of \(A\) are orthonormal

\(\implies\) \(A\) injective

Also \(A\) is square. Since injective automorphisms are epimorphisms, we have:

\(\implies\) \(A\) is an isomorphism

\(\implies\) \(A\) is invertible, and \(A^H = A^{-1}\)

This finished the proof.

4.1 Notes

1. Hence for unitary operators: \[ A^H A = A A^H = I \]

2. Note \[ \begin{aligned} &\quad \quad \ \begin{cases} A \text{ is normal} \\ \text{Spectrum of } A \subseteq \mathbb{S}^1 \subseteq \mathbb{C} \end{cases} \\ &\iff A^H = A^{-1} \\ &\iff \begin{cases} A \text{ is square} \\ A^H A = I \end{cases} \\ &\iff \begin{cases} A \text{ is square} \\ A A^H = I \end{cases} \end{aligned} \]