Linear and multilinear algebra; matrix theory Definition

Conjugate transpose

Definition

Let $A = (a_{ij})$ be an $m \times n$ matrix with complex entries. The conjugate transpose of $A$ is the $n \times m$ matrix $A^*$ with entries $(A^*)_{ij} = \overline{a_{ji}}$ for $1 \le i \le n$ and $1 \le j \le m$ , where $\overline{z}$ is the complex conjugate of $z$ .

Remarks

Other notations are $A^H$ and $A^\dagger$ . For a real matrix, the conjugate transpose is the transpose. The rules $(A^*)^* = A$ and $(AB)^* = B^* A^*$ hold. For the standard inner product on $\C^n$ , $\langle Ax, y \rangle = \langle x, A^* y \rangle$ .

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Matrix definition
Transpose definition
Conjugate transpose definition

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Tag	`0EX045E2BP`
URI	`m2://0EX045E2BP`
Kind	definition
Writing	W1 stated
MSC	15A99, 15B57
Defines	Conjugate transpose
Uses	Transpose

Conjugate transpose

Definition

Remarks

Reading path

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