Question

Classify each of the following matrices according as it is (a) real, (b) symmetric, (c) skew-symmetric, (d) Hermitian, or (e) skew-hermitian, and iden

Matrices
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.
$$\begin{bmatrix}1 & 0&-i \\ 0 & -2 & 4-i \\ i&4+i&3 \end{bmatrix}$$
$$\begin{bmatrix}7 & 0&4 \\ 0 & -2 & 10 \\ 4&10&5 \end{bmatrix}$$

2020-11-09
Step 1
Real matrix $$\Rightarrow A$$ real matrix is a matrix whose elements consist entirely of real numbers.
Symmetric matrix $$\Rightarrow A$$ symmetric matrix is a square matrix that is equal to its transpose.
$$A^T=A$$
Skew symmetric matrix $$\Rightarrow A$$ matrix can be skew symmetric only if it is square. If the transpose of a matrix is equal to the negative of itself, the matrix is said to be skew symmetric.
$$A^T=-A$$
Step 2
Hermitian matrices $$\Rightarrow$$ Hermitian matrices can be understood as the complex extension of real symmetric matrices.
$$A^{\theta}=A$$
skew-Hermitian matrices $$\Rightarrow$$ skew-Hermitian matrices can be understood as the complex versions of real skew-symmetric matrices, or as the matrix analogue of the purely imaginary numbers.
$$A^{\theta}=-A$$
Step 3
(1) Consider the provided question,
Let $$A=\begin{bmatrix}1 & 0 & -i \\ 0 & -2&4-i \\ i&4+i&3 \end{bmatrix}$$
Now, check the given matrix for the above condition.
check for Hermitian matrix,
$$\bar{A}=\begin{bmatrix}1 & 0 & i \\ 0 & -2&4+i \\ -i&4-i&3 \end{bmatrix}$$ Now $$A^{\theta}=\bar{A}^T=\begin{bmatrix}1 & 0 & -i \\ 0 & -2&4-i \\ i&4+i&3 \end{bmatrix}$$
Therefore , $$A^{\theta}=A$$
So, it is satisfy the Hermitian matrix,
Step 4
The principal diagonal element of the matrix,
$$\begin{bmatrix}1 & 0 & -i \\ 0 & -2&4-i \\ i&4+i&3 \end{bmatrix}$$ is $$\begin{bmatrix}1 & -2 & 3 \end{bmatrix}$$
The Secondary diagonal element of the matrix,
$$\begin{bmatrix}1 & 0 & -i \\ 0 & -2&4-i \\ i&4+i&3 \end{bmatrix}$$ is $$\begin{bmatrix}-i & -2 & i \end{bmatrix}$$
Step 5
(2) Consider the provided question,
Let $$A=\begin{bmatrix}7 & 0 & 4 \\ 0 & -2&10 \\ 4&10&5 \end{bmatrix}$$
Now, check the given matrix for the above condition.
check for symmetric matrix,
$$A^T=\begin{bmatrix}7 & 0 & 4 \\ 0 & -2&10 \\ 4&10&5 \end{bmatrix}$$ therefore , $$A^T=A$$
So, it is satisfy the symmetric matrix,
Step 6
The principal diagonal element of the matrix,
$$\begin{bmatrix}7 & 0 & 4 \\ 0 & -2&10 \\ 4&10&5 \end{bmatrix}$$ is \begin{bmatrix}7 & -2 & 5 \end{bmatrix}
The Secondary diagonal element of the matrix,
$$\begin{bmatrix}7 & 0 & 4 \\ 0 & -2&10 \\ 4&10&5 \end{bmatrix}$$ is \begin{bmatrix}4 & -2 & 4 \end{bmatrix}