📂Matrix Algebra

Conjugate Transpose Matrix

Definition

Let $A$ be a complex matrix of size $m \times n $. Define $\overline{A}$ as follows, and call it the conjugate matrix of $A$.

$$ \overline{A} :=\begin{bmatrix} \overline{a_{11}} & \overline{a_{12}} & \cdots & \overline{a_{1n}} \\ \overline{a_{21}} & \overline{a_{22}} & \cdots & \overline{a_{2n}} \\ \vdots & \vdots & \ddots & \vdots \\ \overline{a_{m1}} & \overline{a_{m2}} & \cdots & \overline{a_{mn}} \end{bmatrix} = \left[ \overline{a_{ij}} \right] $$

Here, $\overline{a}$ is the conjugate complex number of $a$. In other words, a matrix whose elements are the conjugate complex numbers of the elements of another matrix is called a conjugate matrix. Let $A$ be a complex matrix of size $m\times n$ again. Define $A^{\ast}$ as follows, and call it the conjugate transpose of $A$.

$$ A^{\ast} := \overline{A^{T}} = \left( \overline{A} \right) ^{T} $$

Description

In addition to $A^{\ast}$, other notations used are $A^{\dagger}$ and $A^{H}$. $A^{\dagger}$ is read as [Adagger], and $H$ of $A^{H}$ comes from the Hermitian matrix. In physics, especially in quantum mechanics, $A^{\ast}$ is sometimes used only in the sense of a conjugate matrix. Thus, it is denoted as $A^{\dagger}=(A^{\ast})^{T}$. Meanwhile, in numerical linear algebra and elsewhere, $A^{\dagger}$ is used as the notation for a ‘pseudoinverse’, which acts like an inverse matrix, though it is not actually one. Given the wide use of linear algebra, resolving such notation issues requires one to stay focused and closely follow the subject being studied at the time.

Properties¹

Let $A,B$ be any complex matrix, and let $k\in \mathbb{C}$.

(a) $\overline{\overline{A}}=A$

(b) $\overline{(AB)} = \overline{A}\ \overline{B}$

(c) $(A^{\ast})^{\ast}=A$

(d) $\left( A \pm B\right)^{\ast} = A^{\ast} \pm B^{\ast}$

(e) $(kA)^{\ast}=\overline{k}A^{\ast}$

(f) $\left( AB \right)^{\ast} = B^{\ast} A^{\ast}$

Proof

(a) (b)

Obvious from the properties of conjugate complex numbers and the definition of matrix multiplication.

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(c) (d) (e)

Proven by (a), properties of the transpose matrix $ \left( A^{T} \right) ^{T} = A $ , and the definition of matrix addition.

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(f)

Proven by (b), properties of the transpose matrix $\left( AB \right) ^{T} = B^{T} A^{T}$.

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