# Define Hermitian Matrices.

Question
Matrices
Define Hermitian Matrices.

2020-10-19
Step 1
Hermitian matrix is a complex square matrix that is equal to its own conjugate transpose, that is for which
$$A=A^H$$
where $$A^H$$ denotes the conjugate transpose.
In other words, we can say the matrix whose matrix whose element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column is called Hermitian matrix.
That is
$$a_{ij}=\bar{a}_{ji}$$
Step 2
For example, consider the matrix
$$A=\begin{bmatrix}1 & -i \\i & 1 \end{bmatrix}$$
$$\bar{A}=\begin{bmatrix}1 & i \\ -i & 1 \end{bmatrix}$$
$$\bar{A}^T=\begin{bmatrix}1 & -i \\i & 1 \end{bmatrix}$$
$$\bar{A}^T=A$$
Therefore, A is hermitian matrix.

### Relevant Questions

Let A and B be Hermitian matrices. Answer true or false for each of the statements that follow. In each case, explain or prove your answer. The eigenvalues of AB are all real.
Classify each of the following matrices according as it is (a) real, (b) symmetric, (c) skew-symmetric, (d) Hermitian, or (e) skew-hermitian, and identify its principal and secondary diagonals.
$$\begin{bmatrix}1 & 0&-i \\ 0 & -2 & 4-i \\ i&4+i&3 \end{bmatrix}$$
$$\begin{bmatrix}7 & 0&4 \\ 0 & -2 & 10 \\ 4&10&5 \end{bmatrix}$$
Define Exponential Matrices?
Define Block Matrices.
Define Scalar multiplication of matrices?
Let M be the vector space of $$2 \times 2$$ real-valued matrices.
$$M=\begin{bmatrix}a & b \\c & d \end{bmatrix}$$
and define $$M^{#}=\begin{bmatrix}d & b \\c & a \end{bmatrix}$$ Characterize the matrices M such that $$M^{#}=M^{-1}$$
$$a=\left[a_1a_2 \dots a_p\right] \text{ and } b=\left[b_1b_2 \dots b_p\right]^T$$