Linear Algebra: Hermitian and Orthogonal Matrices

What is the form in which every Hermitian matrix A can be written?

Given the Hermitian Matrix A = (\begin{pmatrix} 2 - i & 2 \ 2 + i & 3 \ -2i & 1 \end{pmatrix}), what are the characteristics of matrices B and C?

What defines an orthogonal matrix?

What is a property of orthogonal matrices concerning their determinant?

If A is an orthogonal matrix, what can be said about A' and A^{-1}?

If A and B are orthogonal square matrices of order n, what can be said about A'AB and BA?

Check if the matrix (A = \begin{pmatrix} -8 & 2 \ -2 & 2 \ 1 & 2 \end{pmatrix}) is orthogonal.

Is the matrix (xA = \begin{pmatrix} 4 & 1 \ 4 & -8 \ -2 & 2 \end{pmatrix}) orthogonal?

If A = (\begin{pmatrix} 1 & b \ c \end{pmatrix}) is orthogonal, what are the values of a, b, c?

Check if the following matrices are orthogonal and find A^{-1}: (\begin{pmatrix} 1 & \frac{1}{\sqrt{3}} & \cos a \end{pmatrix}), (\begin{pmatrix} \frac{1}{\sqrt{2}} & -2 \end{pmatrix}), (\begin{pmatrix} \frac{1}{\sqrt{3}} & 1 & 1 \end{pmatrix}).

Is the matrix A = (\begin{pmatrix} 12 & -3 \ 4 \end{pmatrix}) orthogonal?

Hermitian Matrices

A Hermitian Matrix A can be expressed as B + iC, where B is a real symmetric matrix and C is a real skew-symmetric matrix.
Example: The Hermitian matrix A =
```
[2-i  2  2+i]
[2   3   -2i]
[2+i  -2i  1] 
```
can be expressed as B + iC, where B is a real symmetric matrix and C is a real skew-symmetric matrix.

Orthogonal Matrices

A real square matrix A is called orthogonal if:
- AA' = A' A = I (where I is the identity matrix)
Properties of Orthogonal Matrices
- The determinant of an orthogonal matrix (A) is 1 or -1 (|A| = ±1)
- The transpose (A') and inverse (A^-1) of an orthogonal matrix are also orthogonal.
- If A and B are two orthogonal square matrices of order n, then:
  - A' A is orthogonal
  - AB is orthogonal
  - BA is orthogonal
  - A^-1 exists and is equal to A'

Exercise 1.2

Check if the matrix A =
```
[-8  2]
[-2  1]
```
is orthogonal and find its inverse (A^-1).

Practice Exercise 1.2

Check if the following matrices are orthogonal and find their inverses (A^-1):

A =
```
[1  √3]
[√2 -2]
[√2  1 -√3]
```

A =

[cos(a)  sin(a)]
[sin(a)  0]
[cos(a)  0]

A =

[√3  √6  √2]
[1  0  1]
[√3  1  1]

Is the matrix A =
```
[-3  4  12]
[12  -3  4]
[4  12  -3]
```
orthogonal? If not, can it be converted into an orthogonal matrix?
If A =
```
[1  b]
[a  c]
```
is orthogonal then find a, b, and c. Also, find A^-1.