Matrix

tags
Linear Algebra, Math

We star with a set of equations with nine coefficients, three unknowns, and three right-hand sides.

\begin{align*} 2u + v + w &= 5 \\\
4u - 6v &=-2 \\\
-2u + 7v + 2w &=9 \end{align*}

We represent the right hand side with a column vector (or a 1 by 3 matrix). The unknowns of the system are represented as a column vector as well.

\[x=\left[\begin{array}{l}{u} \ {v} \ {w}\end{array}\right]\]

We define a square 3 by 3 matrix of the coefficients

\[A=\left[\begin{array}{ccc} {2} & {1} & {1} \\\
{4} & {-6} & {0} \\\
{-2} & {7} & {2} \end{array}\right]\]

While we present a square matrix here, matrices can be rectangular generally with \(m\) rows and \(n\) columns called an \(m\) by \(n\) matrix.

We can then define the equation as the matrix form \(Ax = b\)

\begin{align*} A x &= b \\ \left[\begin{array}{ccc}{2} & {1} & {1} \\ {4} & {-6} & {0} \\ {-2} & {7} & {2}\end{array}\right]\left[\begin{array}{c}{u} \\ {v} \\ {w}\end{array}\right]&=\left[\begin{array}{c}{5} \\ {-2} \\ {9}\end{array}\right] \end{align*}.

Operations

Addition

This is straightforward. We add each coefficient of the lhs to the corresponding coefficient in the rhs.

\begin{align*} \left[\begin{array}{cc} {2} & {1} \\\
{3} & {0} \\\
{0} & {4} \end{array}\right] + \left[\begin{array}{cc} {1} & {2} \\\
{-3} & {1} \\\
{1} & {2} \end{array}\right] = \left[\begin{array}{ll} {3} & {3} \\\
{0} & {1} \\\
{1} & {6} \end{array}\right] \end{align*}

Product

The first straightforward product is the scalar product. This is straightforward and follows from the definition of addition

\[2\mathbf{A} = \mathbf{A} + \mathbf{A}; \{\mathbf{A} \in \mathcal{R}^{m \times n}\}\]

We can also define a product between two matrices \(\mathbf{A}: n \times m\) and \(\mathbf{B}: m \times k\).

\begin{align*} \left[\begin{array}{ccc} {A_{11}} & {A_{12}} & {A_{13}} \\\
{A_{21}} & {A_{22}} & {A_{23}} \end{array}\right] \times \left[\begin{array}{cc} {B_{11}} & {B_{12}} \\\
{B_{21}} & {B_{22}} \\\
{B_{31}} & {B_{32}} \end{array}\right] &= \left[\begin{array}{ll} {\sum_{i=1}^3 A_{1i} B_{i1}} & {\sum_{i=1}^3 A_{1i} B_{i2}} \\\
{\sum_{i=1}^3 A_{2i} B_{i1}} & {\sum_{i=1}^3 A_{2i} B_{i2}} \end{array}\right] \end{align*}

  • Hadamard product

    The Hadamard product is often used in machine learning. It is quite simply the element-wise produce between two matrices: \(\mathcal{A}, \mathcal{B}: n \times m\).

    \begin{align*} \left[\begin{array}{cc} {A_{11}} & {A_{12}} \\\
    {A_{21}} & {A_{22}} \\\
    {A_{31}} & {A_{32}} \end{array}\right] \odot \left[\begin{array}{cc} {B_{11}} & {B_{12}} \\\
    {B_{21}} & {B_{22}} \\\
    {B_{31}} & {B_{32}} \end{array}\right] &= \left[\begin{array}{ll} {A_{11}B_{11}} & {A_{12}B_{12}} \\\
    {A_{21}B_{21}} & {A_{22}B_{22}} \\\
    {A_{31}B_{31}} & {A_{32}B_{32}} \end{array}\right] \\\
    \left[\begin{array}{cc} {2} & {1} \\\
    {3} & {0} \\\
    {0} & {4} \end{array}\right] \odot \left[\begin{array}{cc} {1} & {2} \\\
    {-3} & {1} \\\
    {1} & {2} \end{array}\right] &= \left[\begin{array}{ll} {2} & {2} \\\
    {-9} & {0} \\\
    {0} & {8} \end{array}\right] \end{align*}

  • Kronecker Product

Transpose

Transpose mirrors a matrix along its diagonal starting from the top left corner. \[[\mathbf{A}^\top]_{ji} = \mathbf{A}_{ij}\].

Inverse

The inverse of a matrix follows from the inverse of a scalar (i.e. \(\alpha \alpha^{-1} = 1\)). For a matrix we instead want the product of a matrix and its inverse to be the identity matrix:

\[\mathbf{A}\mathbf{A}^{-1} = \mathbf{I}\]

The inverse for a square matrix exists iff its determinant is not zero \(\text{det}(\mathbf{A}) = |\mathbf{A}| \neq 0\). If this condition is met, we can calculate the inverse using the adjugate

\[\mathbf{A} = \frac{1}{|\mathbf{A}|} \text{adj}(\mathbf{A})\].

See below for the adjugate of a matrix.

  • Moore-penrose Inverse

Trace

The trace of a matrix is the sum of arguments on the main diagonal.

\[\text{tr}(\mathbf{A}) = \sum_{i=1}^n a_{ii}\]

  • Properties:

    • \(\text{tr}(\mathbf{A} + \mathbf{B}) = \text{tr}(\mathbf{A}) + \text{tr}(\mathbf{B})\)
    • \(\text{tr}(\mathbf{A}) = \text{tr}(\mathbf{A}^\top)\)
    • \(\text{tr}(\mathbf{A}^{\top} \mathbf{B}) = \text{tr}(\mathbf{A} \mathbf{B}^\top) = \text{tr}(\mathbf{B}^{\top} \mathbf{A}) = \text{tr}(\mathbf{B} \mathbf{A}^{\top}) =\sum_{i, j} \mathbf{A}_{i j} \mathbf{B}_{i j}\)

Adjugate

The adjugate of a matrix is the transpose of the cofactor matrix. The cofactor matrix is

Determinant

There are four main uses for the determinant of a Matrix defined by a series of equations.

Hadamard product

This is an element wise product between two vectors or matrices.