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# 3.4: Basic Complex and Matrix Operations

Difficulty Level: At Grade Created by: CK-12

## Complex numbers

m-file environments have excellent support for complex numbers. The imaginary unit is denoted by

i

or

(as preferred in Electrical Engineering)

j

.

To create complex variables z1=7+j\begin{align*}z_1 = 7 + j\end{align*} and z2=2ejπ\begin{align*}z_2 = 2e^{j\pi}\end{align*} simply enter

z1 = 7 + j

and

z2 = 2*exp(j*pi)

Table 2 gives an overview of the basic functions for manipulating complex numbers, where z\begin{align*}z\end{align*} is a complex number.

Manipulating complex numbers.
m-file
Re(z)\begin{align*}Re(z)\end{align*}
real(z)
Im(z)\begin{align*}Im(z)\end{align*}
imag(z)
mag(z)\begin{align*}mag(z)\end{align*}
abs(z)
(z)\begin{align*}\angle(z)\end{align*}
angle(z)
z\begin{align*}z^*\end{align*}
conj(z)

## Operations on Matrices

In addition to scalars, m-file environments can operate on matrices. Some common matrix operations are shown in Table 3; in this table,

M

and

N

are matrices.

Common matrix operations.
Operation m-file
MN\begin{align*}M N\end{align*}
M*N
M1\begin{align*}M^{-1}\end{align*}
inv(M)
MT\begin{align*}M^T\end{align*}
M'
det(M)\begin{align*}det(M)\end{align*}
det(M)

Some useful facts:

• The functions
length

and

size

are used to find the dimensions of vectors and matrices, respectively.

• Operations can also be performed on each element of a vector or matrix by proceeding the operator by ".", e.g
.*

,

.^

and

./

.

Example 4

Let A=(1111)\begin{align*}A = \begin{pmatrix}1 & 1\\ 1 & 1\end{pmatrix}\end{align*}. Then

A^2

will return AA=(2222)\begin{align*}AA = \begin{pmatrix}2 & 2\\ 2 & 2\end{pmatrix}\end{align*}, while

A.^2

will return (12121212)=(1111)\begin{align*}\begin{pmatrix}1^2 & 1^2\\ 1^2 & 1^2\end{pmatrix} = \begin{pmatrix}1 & 1\\ 1 & 1\end{pmatrix}\end{align*}.

Example 5

Given a vector

x

,

compute a vector

y

having elements y(n)=1sin(x(n))\begin{align*}y (n) = \tfrac{1}{sin(x(n))}\end{align*}. This can be easily be done with the command

y=1./sin(x)

Note that using

/

in place of

./

would result in the (common) error "Matrix dimensions must agree".

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