Complex Matrix

Real and Imaginary Parts in Matrix Entries

Complex matrices use the same row, column, addition, and multiplication rules as real matrices. The difference is that every entry may have both a real and an imaginary part.

A complex matrix with $m \in \mathbb{N}$ rows and $n \in \mathbb{N}$ columns can be written as

\begin{aligned} 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}} \end{aligned}

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

For example, entries such as $2+3i$ and $-1-4i$ belong to $\mathbb C$ . Because conjugation matters for lengths and angles, a plain transpose is no longer the right operation for every geometric calculation.

Adjoint Matrix

For complex matrices, the geometric counterpart of the transpose is the $adjoint$ , also called the conjugate transpose. It $transposes the matrix and conjugates every entry$ .

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:

\begin{aligned} 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} \end{aligned}

For example,

A=\begin{pmatrix}1+i&2\\-i&3-2i\end{pmatrix}

The entry in row $j$ and column $i$ of $A^H$ is therefore $\overline{a_{ij}}$ .

Properties of Adjoint Matrix

The adjoint $reverses the order of a product$ and conjugates scalars. For sums and products below, the matrix dimensions must be compatible.

Rule	Identity	Why it works
Real entries	$A^H=A^T$	Conjugation leaves real entries unchanged.
Taking the adjoint twice	$(A^H)^H=A$	Both the transpose and conjugation are undone.
Addition	$(A+B)^H=A^H+B^H$	Conjugation distributes over addition.
Scalar multiplication	$(\lambda A)^H=\overline\lambda A^H$	The scalar is conjugated with every entry.
Product	$(AB)^H=B^HA^H$	Transposing swaps the order of row-column matching.

For the one-entry matrix $A=(1+i)$ , compare the conjugate of $iA$ with multiplying the conjugate of $A$ by $i$ :

(iA)^H=\overline{-1+i}=-1-i

Euclidean Norm on Complex Spaces

For $v,w\in\mathbb C^n$ , the standard Hermitian inner product is

\langle v,w\rangle=v^Hw=\sum_{i=1}^n\overline{v_i}w_i

Conjugating the first argument ensures that the inner product of a vector with itself is real and nonnegative.

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}_{\geq0}

The $Euclidean norm$ is calculated as:

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

For $v=(1+i,2-i)^T$ , the calculation is concrete:

\begin{aligned} v^Hv &=|1+i|^2+|2-i|^2 \\ &=2+5 \\ &=7 \end{aligned}

This is why using $v^Tv$ would be wrong in a complex space: without conjugation, the result need not be real or positive.

Properties of Euclidean Norm

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

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

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

These properties show that $\|\cdot\|_2$ satisfies the definition of a norm. Using the adjoint in the Hermitian inner product keeps complex lengths, orthogonality, and projections consistent with their real counterparts.