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.
The conjugate transpose of a matrix is the matrix defined bywhere 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 is switched before or after moving the entry to a different position does not change the final result.
Example Define the matrix Its conjugate isand its conjugate transpose is
Several different symbols are used in the literature as alternatives to the symbol we have used thus far.
The most common alternatives are the symbol (for Hermitian):
and the dagger:
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 and such that the operations below are well-defined and any scalar , we have that
provided is a square invertible matrix
A matrix that is equal to its conjugate transpose is called Hermitian (or self-adjoint). In other words, is Hermitian if and only if
Example Consider the matrix Then its conjugate transpose isAs a consequence is Hermitian.
Denote by the -th entry of and by the -th entry of . By the definition of conjugate transpose, we have
Therefore, is Hermitian if and only iffor every and , which also implies that the diagonal entries of must be real: their complex part must be zero in order to satisfy
Below you can find some exercises with explained solutions.
Let the vector be defined by
Compute the product
The conjugate transpose of is
and the product is
Let the matrix be defined by
Compute its conjugate transpose.
We have that
Please cite as:
Taboga, Marco (2021). "Conjugate transpose", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/conjugate-transpose.
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