# Special Matrices & Vectors

## Diagonal Matrix

For such matrices, only diagonal entries are non-zero.

$$
A\_{ij} = 0 \hspace{0.2cm} if \hspace{0.1cm} i\neq j
$$

## ​Symmetric Matrix

Symmetric matrices are equal to their transpose.

$$
A = A^T
$$

## Unit Vector

Vector with unit "length".

$$
||v||\_2 = 1
$$

For example, both $$\begin{pmatrix} 0 \ 0 \ 1 \end{pmatrix}$$​and $$\begin{pmatrix} \frac{1}{\sqrt2} \ \frac{1}{\sqrt2} \ 0 \end{pmatrix}$$​are unit vectors.

## Unit Basis Vector

An important set of vectors is the set of unit basis vectors given by:

$$
e\_j = \begin{pmatrix}
0 \\
0 \\
0 \\
0 \\
... \\
1 \\
0 \\
0 \\
...\\
0 \\
\end{pmatrix}
$$

Here, the “1” appears as the component indexed by j. Thus, we get the set $${e\_0, e\_1, ..., e\_{n-1}} \subset \mathbb{R}^{n}$$given by:

$$
e\_0 = \begin{pmatrix}
1 \\
0 \\
... \\
0 \\
0 \\
\end{pmatrix},
e\_1 = \begin{pmatrix}
0 \\
1 \\
... \\
0 \\
0 \\
\end{pmatrix},
e\_{n-1} = \begin{pmatrix}
0 \\
0 \\
... \\
0 \\
1 \\
\end{pmatrix}
$$

These vectors are also referred to as the **standard basis vectors**.

## Orthogonal Vectors​

Two vectors are called orthogonal to each other if they are mutually prependicual vectors.

$$
x^Ty = 0 \hspace{0.3cm} (Dot \hspace{0.1cm} Product)
$$

​Also, *orthonormal vectors* are unit vectors prependicular to each other.

## Orthogonal Matrix

A matrix is orthogonal if its transpose is equal to its inverse.

$$
A^T = A^{-1}
$$

​Identity matrix is a orthogonal matrix. Also, all columns are orthonormal.

Further, orthogonal matrices represent rotational operations which preserve volume.