What is a Unitary Matrix?
A Unitary Matrix is a form of a complex square matrix in which its conjugate transpose is also its inverse. This means that a matrix is flipped over its diagonal row and the conjugate of its inverse is calculated. If the resulting output, called the conjugate transpose is equal to the inverse of the initial matrix, then it is unitary. Unitary matrices have a few properties specific to their form. For example, a unitary matrix U
must be normal, meaning that, when multiplying by its conjugate transpose, the order of operations does not affect the result (i.e. XY=YX). Similarly, U
must be diagonalizable meaning its form is unitarily similar to a diagonal matrix, in which all values aside from the main diagonal are zero. Furthermore, a unitary matrix' eigenspaces must be orthogonal. This means that the values in which the matrix does not change, must also be orthogonal.
How does a Unitary Matrix work?
Let's break down the definition a little more to understand the form and function of unitary matrices. As mentioned above, a unitary matrix' conjugate transpose is also its inverse. In mathematics, a conjugate transpose is the process of taking the transpose of a matrix, meaning flip it over its main diagonal, and then finding the complex conjugate. A complex conjugate of a number is the number with an equal real part and imaginary part, equal in magnitude, but opposite in sign. For example, the complex conjugate of X+iY is X-iY. If the conjugate transpose of a square matrix is equal to its inverse, then it is a unitary matrix.