Complex Matrix

Definition of Matrix with Complex Entries

Just like matrices with real entries, we can also form matrices whose entries are complex numbers. Imagine a rectangular table containing complex numbers arranged neatly in rows and columns.

A rectangular scheme of complex numbers with $m \in \mathbb{N}$ rows and $n \in \mathbb{N}$ columns is called a complex $m \times n$ matrix:

A = \begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & & \vdots \\ a_{m1} & \cdots & a_{mn} \end{pmatrix} = (a_{ij})_{\substack{i=1,\ldots,m \\ j=1,\ldots,n}}

with coefficients $a_{ij} \in \mathbb{C}$ for $i = 1, \ldots, m$ and $j = 1, \ldots, n$ . The set of all complex $m \times n$ matrices is written as $\mathbb{C}^{m \times n}$ .

The main difference from real matrices is that each entry $a_{ij}$ can now be a complex number such as $2 + 3i$ or $-1 - 4i$ .

Adjoint Matrix

In the context of complex matrices, the concept of matrix transpose is extended to a more general concept called the adjoint matrix. This concept is the complex generalization of the transpose matrix $A^T$ .

Let $A \in \mathbb{C}^{m \times n}$ be a complex $m \times n$ matrix. The adjoint matrix (complex conjugate transpose matrix) $A^H$ of $A$ is the complex $n \times m$ matrix obtained by swapping the rows and columns of $A$ and taking the complex conjugate of each entry:

A^H = \overline{A^T} = \overline{A}^T = \begin{pmatrix} \overline{a_{11}} & \cdots & \overline{a_{m1}} \\ \vdots & & \vdots \\ \overline{a_{1n}} & \cdots & \overline{a_{mn}} \end{pmatrix} \in \mathbb{C}^{n \times m}

This process involves two steps: first transpose the matrix (swap rows and columns), then take the complex conjugate of each entry.

Properties of Adjoint Matrix

The adjoint matrix has several important properties that are useful in calculations. Here are the fundamental properties that always hold:

Basic Properties

Relationship with real transpose: For $A \in \mathbb{R}^{m \times n} \subset \mathbb{C}^{m \times n}$ , we have $A^H = A^T$

This makes sense because if the matrix only contains real entries, then the complex conjugate does not change its value.

Involution: $(A^H)^H = A$

If we take the adjoint of an adjoint, we return to the original matrix.

Linear Operation Properties

Linearity of addition: $(A + B)^H = A^H + B^H$
Linearity of scalar multiplication: $(\lambda \cdot A)^H = \overline{\lambda} \cdot A^H$

Note that for scalar multiplication, we need to take the conjugate of the scalar $\lambda$ .

Multiplication Properties

Anticommutative property of multiplication: $(A \cdot B)^H = B^H \cdot A^H$

This property shows that the adjoint of a matrix product is the product of the adjoints of those matrices in reverse order.

Euclidean Norm on Complex Spaces

For vectors in complex spaces, we need a way to measure the "length" or norm of that vector. The concept of Euclidean norm is extended for complex spaces using the adjoint matrix.

For $v \in \mathbb{C}^n$ , we define:

v^H v = \sum_{i=1}^n \overline{v_i} v_i = \sum_{i=1}^n |v_i|^2 \in \mathbb{R}_0^+

The Euclidean norm is calculated as:

\|v\|_2 = \sqrt{v^H v}

This defines a norm on $\mathbb{C}^n$ , namely the Euclidean norm $\|\cdot\|_2 : \mathbb{C}^n \to \mathbb{R} : v \mapsto \|v\|_2$ .

Properties of Euclidean Norm

The Euclidean norm satisfies the following properties:

Homogeneity: $\|\alpha v\|_2 = |\alpha| \|v\|_2$

Positive definite: $\|v\|_2 \geq 0$ and $\|v\|_2 = 0 \Leftrightarrow v = 0$

Triangle inequality: $\|v + w\|_2 \leq \|v\|_2 + \|w\|_2$

These properties guarantee that $\|\cdot\|_2$ is truly a norm in the formal mathematical sense.

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