0.1 A matrix times a vector

 Page 1 / 5
One can look at the operation of a matrix times a vector as changing the basis set for the vector or as changing the vector with the same basis description. Many signal and systems problem can be posed in this form.

A matrix times a vector

In this chapter we consider the first problem posed in the introduction

$\mathbf{A}\mathbf{x}=\mathbf{b}$

where the matrix $\mathbf{A}$ and vector $\mathbf{x}$ are given and we want to interpret and give structure to the calculation of the vector $\mathbf{b}$ . Equation [link] has a variety of special cases. The matrix $\mathbf{A}$ may be square or may be rectangular. It may have full column or row rank or it may not. It may be symmetric or orthogonalor non-singular or many other characteristics which would be interesting properties as an operator. If we view the vectors as signals and thematrix as an operator or processor, there are two interesting interpretations.

• The operation [link] is a change of basis or coordinates for a fixed signal. The signal stays the same, the basis (or frame) changes.
• The operation [link] alters the characteristics of the signal (processes it) but within a fixed basis system. The basis stays the same, the signalchanges.

An example of the first would be the discrete Fourier transform (DFT) where one calculates frequency components of a signal which arecoordinates in a frequency space for a given signal. The definition of the DFT from [link] can be written as a matrix-vector operation by $\mathbf{c}=\mathbf{Wx}$ which, for $w={e}^{-j2\pi /N}$ and $N=4$ , is

$\left[\begin{array}{c}{c}_{0}\\ {c}_{1}\\ {c}_{2}\\ {c}_{3}\end{array}\right]=\left[\begin{array}{cccc}{w}^{0}& {w}^{0}& {w}^{0}& {w}^{0}\\ {w}^{0}& {w}^{1}& {w}^{2}& {w}^{3}\\ {w}^{0}& {w}^{2}& {w}^{4}& {w}^{6}\\ {w}^{0}& {w}^{3}& {w}^{6}& {w}^{9}\end{array}\right]\left[\begin{array}{c}{x}_{0}\\ {x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right]$

An example of the second might be convolution where you are processing or filtering asignal and staying in the same space or coordinate system.

$\left[\begin{array}{c}{y}_{0}\\ {y}_{1}\\ {y}_{2}\\ ⋮\end{array}\right]=\left[\begin{array}{ccccc}{h}_{0}& 0& 0& \cdots & 0\\ {h}_{1}& {h}_{0}& 0& & \\ {h}_{2}& {h}_{1}& {h}_{0}& & \\ ⋮& & & & ⋮\end{array}\right]\left[\begin{array}{c}{x}_{0}\\ {x}_{1}\\ {x}_{2}\\ ⋮\end{array}\right].$

A particularly powerful sequence of operations is to first change the basis for a signal, then process the signal in this new basis, and finally returnto the original basis. For example, the discrete Fourier transform (DFT) of a signal is taken followed by setting some of the Fourier coefficients tozero followed by taking the inverse DFT.

Another application of [link] is made in linear regression where the input signals are rows of $\mathbf{A}$ and the unknown weights of the hypothesisare in $\mathbf{x}$ and the outputs are the elements of $\mathbf{b}$ .

Change of basis

Consider the two views:

1. The operation given in [link] can be viewed as $\mathbf{x}$ being a set of weights so that $\mathbf{b}$ is a weighted sum of the columns of $\mathbf{A}$ . In other words, $\mathbf{b}$ will lie in the space spanned by the columns of $\mathbf{A}$ at a location determined by $\mathbf{x}$ . This view is a composition of a signal from a set of weights as in [link] and [link] below. If the vector ${\mathbf{a}}_{\mathbf{i}}$ is the ${i}^{th}$ column of $\mathbf{A}$ , it is illustrated by
$\mathbf{A}\mathbf{x}={x}_{1}\left[\begin{array}{c}⋮\\ {\mathbf{a}}_{\mathbf{1}}\\ ⋮\end{array}\right]+{x}_{2}\left[\begin{array}{c}⋮\\ {\mathbf{a}}_{\mathbf{2}}\\ ⋮\end{array}\right]+{x}_{3}\left[\begin{array}{c}⋮\\ {\mathbf{a}}_{\mathbf{3}}\\ ⋮\end{array}\right]=\mathbf{b}.$
2. An alternative view has $\mathbf{x}$ being a signal vector and with $\mathbf{b}$ being a vector whose entries are inner products of $\mathbf{x}$ and the rows of A . In other words, the elements of $\mathbf{b}$ are the projection coefficients of $\mathbf{x}$ onto the coordinates given by the rows of A . The multiplication of a signal by this operator decomposes the signal and gives thecoefficients of the decomposition. If ${\overline{\mathbf{a}}}_{\mathbf{j}}$ is the ${j}^{th}$ row of $\mathbf{A}$ we have:
${b}_{1}=\left[\begin{array}{c}\cdots {\overline{\mathbf{a}}}_{\mathbf{1}}\cdots \end{array}\right]\left[\begin{array}{c}⋮\\ \mathbf{x}\\ ⋮\end{array}\right]\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{b}_{2}=\left[\begin{array}{c}\cdots {\overline{\mathbf{a}}}_{\mathbf{2}}\cdots \end{array}\right]\left[\begin{array}{c}⋮\\ \mathbf{x}\\ ⋮\end{array}\right]\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}etc.$
Regression can be posed from this view with the input signal being the rows of A .

what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!