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.