Answer to Question #102640 in Linear Algebra for BIVEK SAH

Question

Define skew-hermitian matrix, orthogonal matrix and unitary matrix with an
example of each.

Answer 1

A square matrix with complex entries is said to be skew hermitian if its conjugate transpose is the negative of the original matrix.

or,$A=-A^H$ where ${\displaystyle A^{\textsf {H}}}$  denotes the conjugate transpose of the matrix ${\displaystyle A}$ .

Example:

$\begin{bmatrix} i & 1+i \\ -1+i & 3i \end{bmatrix}$ as $A_{ij}=-\overline{A_{ji}}$

A square matrix with real numbers or elements is said to be an orthogonal matrix, if its transpose is equal to its inverse matrix or we can say, when the product of a square matrix and its transpose gives an identity matrix, then the square matrix  is known as orthogonal matrix.

or,$AA^{T}=I$ where $A$ is a square matrix and $I$ is identity matrix of same order.

Example $-$ $\begin{bmatrix} cos Z & sin Z \\ -sinZ & cosZ \end{bmatrix}$

The determinant of a orthogonal matrix is $+1$ or $-1$ .

A unitary matrix is a matrix whose inverse equals it conjugate transpose. Unitary matrices are the complex analog of real orthogonal matrices.

OR, 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. 

Properties.

1. 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.$XY=YX$ ).
2. $U$ must be diagonalizable meaning its form is unitarily similar to a diagonal matrix, in which all values aside from the main diagonal are zero.
3.  A unitary matrix' eigenspaces must be orthogonal. This means that the values in which the matrix does not change, must also be orthogonal.

Example.

$\begin{bmatrix} 1+i & 1-i \\ 1-i & 1+i \end{bmatrix}$