Search for probability and statistics terms on Statlect

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


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]


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]


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

Hermitian matrix

A matrix that is equal to its conjugate transpose is called Hermitian (or self-adjoint). In other words, A is Hermitian if and only if[eq12]

Example Consider the matrix [eq13]Then its conjugate transpose is[eq14]As a consequence A is Hermitian.

Denote by $A_{kl}$ the $left( k,l
ight) $-th entry of A and by [eq15] the $left( k,l
ight) $-th entry of $A^{st }$. By the definition of conjugate transpose, we have[eq16]

Therefore, A is Hermitian if and only if[eq17]for every k and $l$, which also implies that the diagonal entries of A must be real: their complex part must be zero in order to satisfy[eq18]

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Let the vector A be defined by[eq19]

Compute the product[eq20]


The conjugate transpose of A is[eq21]

and the product is[eq22]

Exercise 2

Let the matrix A be defined by[eq23]

Compute its conjugate transpose.


We have that[eq24]

How to cite

Please cite as:

Taboga, Marco (2021). "Conjugate transpose", Lectures on matrix algebra.

The books

Most of the learning materials found on this website are now available in a traditional textbook format.