Unitary Matrix

Similarly, a matrix $U$ is ==unitary== if its conjugate transpose is equal to its inverse ( $U^{\ast }=U^{-1}$ ).

This is similar for orthogonal, in linear algebra, a matrix $A$ is orthogonal if its transpose is equal to its inverse ( $A^T=A^{-1}$ ).

These properties are essential in various applications, including solving systems of equations, transforming coordinates, and diagonalizing matrices.

In terms of individual matrix elements, if $U=[u_{ij}]$, then the condition for unitarity is:

$\sum (u_{ik}^{\ast }\times u_{jk}^{\ast })={\delta }_{ij}$

Where ${\delta }_{ij}$ is Kronecker delta, which is 1 when $i=j$ and 0 otherwise.

Unitary matrices preserve the inner product and length of vectors when applied to them.

Geometrically, they represent rotations and reflections in a complex vector space. In quantum mechanics, unitary matrices are crucial for describing time evolution of quantum states through unitary transformations.

Properties of unitary matrices include:

• The determinant of a unitary matrix has a magnitude of 1: $|\det (U)|=1$.
• The eigenvalues of a unitary matrix lie on the unit circle in the complex plane.
• The columns (and rows) of a unitary matrix form an orthonormal basis.

Common examples of unitary matrices include rotation matrices in two- and three-dimensional spaces, as well as certain quantum gates in quantum computing.

In summary, unitary matrices are a specialized class of matrices that maintain certain important properties while transforming vectors and are particularly useful in fields dealing with complex numbers, linear transformations, and quantum mechanics.