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
ANSWERED
asked 2020-11-08
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.
\(\begin{bmatrix}1 & 0&-i \\ 0 & -2 & 4-i \\ i&4+i&3 \end{bmatrix}\)
\(\begin{bmatrix}7 & 0&4 \\ 0 & -2 & 10 \\ 4&10&5 \end{bmatrix}\)

Answers (1)

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}
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