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URL: https://nakafa.com/en/subjects/ai-ds/linear-methods/complex-matrix
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Learn complex matrices with adjoint operations, Hermitian transpose, Euclidean norms, and essential properties for quantum computing and signal processing.

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## 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:

Visible text: A rectangular scheme of complex numbers with rows and columns is called a complex matrix:

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

Visible text: with **coefficients** for and . The set of all complex matrices is written as .

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

Visible text: The main difference from real matrices is that each entry can now be a complex number such as or .

## 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 $$A^T$$.

Visible text: 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 .

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:

Visible text: Let be a complex matrix. The **adjoint matrix** (complex conjugate transpose matrix) of is the complex matrix obtained by swapping the rows and columns of and taking the complex conjugate of each entry:

```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 $$A \in \mathbb{R}^{m \times n} \subset \mathbb{C}^{m \times n}$$, we have $$A^H = A^T$$

Visible text: 1. **Relationship with real transpose**: For , we have

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

2. **Involution**: $$(A^H)^H = A$$

Visible text: 2. **Involution**:

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

### Linear Operation Properties

3. **Linearity of addition**: $$(A + B)^H = A^H + B^H$$

4. **Linearity of scalar multiplication**: $$(\lambda \cdot A)^H = \overline{\lambda} \cdot A^H$$

Visible text: 3. **Linearity of addition**: 

4. **Linearity of scalar multiplication**:

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

Visible text: Note that for scalar multiplication, we need to take the conjugate of the scalar .

### Multiplication Properties

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

Visible text: 5. **Anticommutative property of multiplication**:

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:

Visible text: For , we define:

```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:

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

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

Visible text: This defines a norm on , namely the **Euclidean norm** .

### Properties of Euclidean Norm

The Euclidean norm satisfies the following properties:

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

Visible text: **Homogeneity**:

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

Visible text: **Positive definite**: and

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

Visible text: **Triangle inequality**:

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

Visible text: These properties guarantee that is truly a norm in the formal mathematical sense.