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Conjugate transpose

by , PhD

It often happens in matrix algebra that we need to both transpose and take the complex conjugate of a matrix. The result of the sequential application of these two operations is called conjugate transpose (or Hermitian transpose). Special symbols are used in the mathematics literature to denote this double operation.

Table of Contents

Definition

The conjugate transpose of a matrix A is the matrix $A^{st }$ defined by[eq1]where $	op $ denotes transposition and the over-line denotes complex conjugation.

Remember that the complex conjugate of a matrix is obtained by taking the complex conjugate of each of its entries (see the lecture on complex matrices).

In the definition we have used the fact that the order in which transposition and conjugation are performed is irrelevant: whether the sign of the imaginary part of an entry of A is switched before or after moving the entry to a different position does not change the final result.

Example Define the matrix [eq2]Its conjugate is[eq3]and its conjugate transpose is[eq4]

Symbols

Several different symbols are used in the literature as alternatives to the $st $ symbol we have used thus far.

The most common alternatives are the H symbol (for Hermitian):[eq5]

and the dagger:[eq6]

Properties

The properties of conjugate transposition are immediate consequences of the properties of transposition and conjugation. We therefore list some of them without proofs.

For any two matrices A and $B$ such that the operations below are well-defined and any scalar $zin U{2102} $, we have that

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Let the vector A be defined by[eq12]

Compute the product[eq13]

Solution

The conjugate transpose of A is[eq14]

and the product is[eq15]

Exercise 2

Let the matrix A be defined by[eq16]

Compute its conjugate transpose.

Solution

We have that[eq17]

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