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Special Matrices & Vectors

Diagonal Matrix

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

Aij=0ifijA_{ij} = 0 \hspace{0.2cm} if \hspace{0.1cm} i\neq j

​Symmetric Matrix

Symmetric matrices are equal to their transpose.

A=ATA = A^T

Unit Vector

Vector with unit "length".

v2=1||v||_2 = 1

For example, both (001)\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}​and (12120)\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:

ej=(0000...100...0)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 {e0,e1,...,en1}Rn\{e_0, e_1, ..., e_{n-1}\} \subset \mathbb{R}^{n}given by:

e0=(10...00),e1=(01...00),en1=(00...01)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.

xTy=0(DotProduct)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.

AT=A1A^T = A^{-1}

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

Further, orthogonal matrices represent rotational operations which preserve volume.

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