# Nakafa Framework: LLM

URL: /en/subject/university/bachelor/ai-ds/linear-methods/complex-matrix
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/university/bachelor/ai-ds/linear-methods/complex-matrix/en.mdx

Output docs content for large language models.

---

export const metadata = {
    title: "Complex Matrix",
    description: "Learn complex matrices with adjoint operations, Hermitian transpose, Euclidean norms, and essential properties for quantum computing and signal processing.",
    authors: [{ name: "Nabil Akbarazzima Fatih" }],
    date: "07/12/2025",
    subject: "Linear Methods of AI",
};

## 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 <InlineMath math="m \in \mathbb{N}" /> rows and <InlineMath math="n \in \mathbb{N}" /> columns is called a complex <InlineMath math="m \times n" /> matrix:

<BlockMath math="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** <InlineMath math="a_{ij} \in \mathbb{C}" /> for <InlineMath math="i = 1, \ldots, m" /> and <InlineMath math="j = 1, \ldots, n" />. The set of all complex <InlineMath math="m \times n" /> matrices is written as <InlineMath math="\mathbb{C}^{m \times n}" />.

The main difference from real matrices is that each entry <InlineMath math="a_{ij}" /> can now be a complex number such as <InlineMath math="2 + 3i" /> or <InlineMath math="-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 <InlineMath math="A^T" />.

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

<BlockMath math="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

1. **Relationship with real transpose**: For <InlineMath math="A \in \mathbb{R}^{m \times n} \subset \mathbb{C}^{m \times n}" />, we have <InlineMath math="A^H = A^T" />

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

2. **Involution**: <InlineMath math="(A^H)^H = A" />

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

### Linear Operation Properties

3. **Linearity of addition**: <InlineMath math="(A + B)^H = A^H + B^H" />

4. **Linearity of scalar multiplication**: <InlineMath math="(\lambda \cdot A)^H = \overline{\lambda} \cdot A^H" />

Note that for scalar multiplication, we need to take the conjugate of the scalar <InlineMath math="\lambda" />.

### Multiplication Properties

5. **Anticommutative property of multiplication**: <InlineMath math="(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 <InlineMath math="v \in \mathbb{C}^n" />, we define:

<BlockMath math="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:

<BlockMath math="\|v\|_2 = \sqrt{v^H v}" />

This defines a norm on <InlineMath math="\mathbb{C}^n" />, namely the **Euclidean norm** <InlineMath math="\|\cdot\|_2 : \mathbb{C}^n \to \mathbb{R} : v \mapsto \|v\|_2" />.

### Properties of Euclidean Norm

The Euclidean norm satisfies the following properties:

**Homogeneity**: <InlineMath math="\|\alpha v\|_2 = |\alpha| \|v\|_2" />

**Positive definite**: <InlineMath math="\|v\|_2 \geq 0" /> and <InlineMath math="\|v\|_2 = 0 \Leftrightarrow v = 0" />

**Triangle inequality**: <InlineMath math="\|v + w\|_2 \leq \|v\|_2 + \|w\|_2" />

These properties guarantee that <InlineMath math="\|\cdot\|_2" /> is truly a norm in the formal mathematical sense.