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# Matrices basics

Modified 2018-06-22 by Andrea Censi

Dzenan Lapandic

k:basic_math

k:linear_algebra

k:matrices

A matrix:

$$\amat{A} = \left[ \begin{array}{ccc} a_{11} & \dots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \dots & a_{mn} \end{array} \right] \in \reals^{m \times n} \label{eq:matrix}\tag{1}$$

is a table ordered by ($m$) horizontal rows and ($n$) vertical columns. Its elements are typically denoted with lower case latin letters, with subscripts indicating their row and column respectively. For example, $a_{ij}$ is the element of $A$ at the $i$-th row and $j$-th column.

A vector is a matrix with one column.

## Matrix dimensions

Modified 2018-06-22 by Andrea Censi

The number of rows and columns of a matrix are referred to as the matrix dimensions. $\amat{A} \in \reals^{m \times n}$ has dimensions $m$ and $n$.

Fat matrix When $n \gt{} m$, i.e., the matrix has more columns than rows, $\amat{A}$ is called fat matrix.

Tall matrix When $n \lt{} m$, i.e., the matrix has more rows than columns, $\amat{A}$ is called tall matrix.

Fat matrix When $n = m$, $\amat{A}$ is called square matrix.

Square matrices are particularly important.

## Diagonal matrix

Modified 2018-06-22 by Andrea Censi

Diagonal matrix A diagonal matrix has non zero elements only on its main diagonal. \begin{align} \amat{A} = \left[ \begin{array}{ccc} a_11 & 0 & \dots & 0 \\ 0 & a_22 & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \dots & 0 & a_nn \end{array} \right] \end{align}

## Identity matrix

Modified 2018-06-22 by Andrea Censi

Identity matrix An identity matrix is a diagonal square matrix with all elements equal to one. \begin{align} \amat{I} = \left[ \begin{array}{ccc} 1 & 0 & \dots & 0 \\ 0 & 1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \dots & 0 & 1 \end{array} \right] \end{align}

## Null matrix

Modified 2018-06-22 by Andrea Censi

Null matrix The null, or Zero, matrix is a matrix whos elements are all zeros. \begin{align} \amat{0} = \left[ \begin{array}{ccc} 0 & \dots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \dots & 0 \end{array} \right] \end{align}

## Determinant

Modified 2018-06-22 by Andrea Censi

• 2x2
• 3x3
• nxn

## Rank of a matrix

Modified 2018-06-22 by Andrea Censi

## Trace of a matrix

Modified 2018-06-22 by Andrea Censi

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