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}