# 15.11 Orthonormal bases in real and complex spaces

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This module defines the terms transpose, inner product, and Hermitian transpose and their use in finding an orthonormal basis.

## Notation

Transpose operator $A^T$ flips the matrix across it's diagonal. $A=\begin{pmatrix}a_{1, 1} & a_{1, 2}\\ a_{2, 1} & a_{2, 2}\\ \end{pmatrix}$ $A^T=\begin{pmatrix}a_{1, 1} & a_{2, 1}\\ a_{1, 2} & a_{2, 2}\\ \end{pmatrix}$ Column $i$ of $A$ is row $i$ of $A^T$

Recall, inner product $x=\begin{pmatrix}{x}_{0}\\ {x}_{1}\\ ⋮\\ {x}_{n-1}\\ \end{pmatrix}$ $y=\begin{pmatrix}{y}_{0}\\ {y}_{1}\\ ⋮\\ {y}_{n-1}\\ \end{pmatrix}$ $x^Ty=\begin{pmatrix}{x}_{0} & {x}_{1} & \dots & {x}_{n-1}\\ \end{pmatrix}\begin{pmatrix}{y}_{0}\\ {y}_{1}\\ ⋮\\ {y}_{n-1}\\ \end{pmatrix}=\sum x_{i}y_{i}=y\dot x$ on $\mathbb{R}^{n}$

Hermitian transpose $(A)$ , transpose and conjugate $(A)=\overline{A^T}$ $y\dot x=(x)y=\sum x_{i}\overline{y_{i}}$ on $\mathbb{C}^{n}$

Now, let $\{{b}_{0}, {b}_{1}, \dots , {b}_{n-1}\}$ be an orthonormal basis for $\mathbb{C}^{n}$ $\forall i, \colon {b}_{i}\dot {b}_{i}=1$ $(i\neq j, {b}_{i}\dot {b}_{j}=({b}_{j}){b}_{i}=0)$

Basis matrix: $B=\begin{pmatrix}⋮ & ⋮ & & ⋮\\ {b}_{0} & {b}_{1} & \dots & {b}_{n-1}\\ ⋮ & ⋮ & & ⋮\\ \end{pmatrix}$ Now, $(B)B=\begin{pmatrix}\dots & ({b}_{0}) & \dots \\ \dots & ({b}_{1}) & \dots \\ & ⋮ & \\ \dots & ({b}_{n-1}) & \dots \\ \end{pmatrix}\begin{pmatrix}⋮ & ⋮ & & ⋮\\ {b}_{0} & {b}_{1} & \dots & {b}_{n-1}\\ ⋮ & ⋮ & & ⋮\\ \end{pmatrix}=\begin{pmatrix}({b}_{0}){b}_{0} & ({b}_{0}){b}_{1} & \dots & ({b}_{0}){b}_{n-1}\\ ({b}_{1}){b}_{0} & ({b}_{1}){b}_{1} & \dots & ({b}_{1}){b}_{n-1}\\ ⋮\\ ({b}_{n-1}){b}_{0} & ({b}_{n-1}){b}_{1} & \dots & ({b}_{n-1}){b}_{n-1}\\ \end{pmatrix}$

For orthonormal basis with basis matrix $B$ $(B)=B^{(-1)}$ ( $B^T=B^{(-1)}$ in $\mathbb{R}^{n}$ ) $(B)$ is easy to calculate while $B^{(-1)}$ is hard to calculate.

So, to find $\{{\alpha }_{0}, {\alpha }_{1}, \dots , {\alpha }_{n-1}\}$ such that $x=\sum {\alpha }_{i}{b}_{i}$ Calculate $(\alpha =B^{(-1)}x)\implies (\alpha =(B)x)$ Using an orthonormal basis we rid ourselves of the inverse operation.

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
absolutely yes
Daniel
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characteristics of micro business
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for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
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
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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 ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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 By By Nick Swain By By By Mldelatte