Matrix Algebra

Matrix Equations

Matlab/Octave. Provided an $m\times n$ matrix A and a vector u in $\mathbb{R}^n$ , we can compute the multiplication as follows.

> A * u

EXAMPLE 1. Compute $A \mathbf{u}$ in each of the following.

$A = \left[\begin{array}{ccc} 1 & 2 & -1 \\ 0 &-5 & 3 \end{array}\right]$ and $\quad \mathbf{u} = \left[\begin{array}{c} 4 \\ 3 \\ 7 \end{array}\right]$
$A = \left[\begin{array}{cc} 2 &-3 \\ 8 & 0 \\ -5 & 2 \end{array}\right]$ and $\quad \mathbf{u} = \left[\begin{array}{c} 4 \\ 7 \end{array}\right]$

Matrix Equations. Consider an $m\times n$ matrix as coefficient to a system of linear equations. Then the product $A \mathbf{x}$ of an $m\times n$ matrix with a -dimensional vector is given by

$\displaystyle \left[\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n} \\ ... ...ots \\ a_{m1} x_1 + a_{m2} x_2 + \cdots + a_{mn} x_n \end{array}\right] ,$

and results in an

-dimensional vector. With another

-dimensional vector $\mathbf{b}$ we can express a system of linear equations as follows.

$\displaystyle \left[\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n} \\ ... ...] = \left[\begin{array}{c} b_1 \\ \vdots \\ b_m \end{array}\right]$

Or, we can simply write $A \mathbf{x} = \mathbf{b}$ .