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.

This quiz covers the properties and representations of Hermitian and orthogonal matrices. It includes examples and exercises to help you understand the concepts better. Test your knowledge on how these matrices are defined and their characteristics in linear algebra.