For questions related to orthonormality. A set of vectors in an inner product space is called orthonormal if each vector is a unit vector, and vectors are pairwise orthogonal.

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Show that the continuum of elements $e^{i\lambda t}$ forms a complete orthonormal subset of $B^2$.

Let $X$ be the vector space of all finite linear combinations of functions of the form $e^{i\lambda t}$ ($-\infty<t<+\infty$), where the parameter $\lambda$ is real. An inner product in ...
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21 views

Gram–Schmidt algorithm used for obtaining the orthogonal and orthonormal

Why are both the algorithm used for finding the orthogonal and orthonormal basis the same? I'm relying on a set of slides given by by lecturer (known to be sloppy!) and I want to confirm if it should ...
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1answer
26 views

What exactly is pairwise orthogonal?

Suppose there exists a basis $$B = \left \{ v_{1},...,v_{n}\right \}$$ and basis $$B' = \left \{ v_{1}',...,v_{n}'\right \}$$ Then, if $$\left \langle B,B' \right \rangle=0$$ then B and B' are ...
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33 views

Simple proof that $\min \|\mathbf{f} - \sum_i \lambda_i\hat{\mathbf{g}_i}\|$ is given by $\lambda_i = \langle \mathbf{f}, \hat{\mathbf{g}}_i \rangle$

Let $\mathbf{f}$ be a real valued vector and $\{\hat{\mathbf{g}}_i\}$ be a set of orthonormal vectors of the same dimension. What is a simple proof that $$\min \|\mathbf{f} - \sum_i ...
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17 views

$\lim_{k \to \infty} \langle s_k,e_n \rangle = \langle h,e_n \rangle$

I took a passage from a textbook regarding equivalent conditions of having an orthonormal sequence in a Hilbert space H. Why is the equality $$\lim_{k \to \infty} \langle s_k,e_n \rangle = \langle ...
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2answers
68 views

Equality of two matrices

If we have a diagonal matrix D which verify $D = A^*MA = B^*MB$ where $^*$ denotes the conjugate transpose, with A, B and M being unitary matrices Plus, B is symetric, M is real, and none of these ...
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29 views

Gram-Schmidt procedure on complex waveforms

I am trying to derive an orthonormal basis for the following complex wave forms using Gram-Schmidt procedure $$s_1(t)=\sqrt{2E\over T}e^{j\phi{[n-1]}} e^{j\pi t\over 2T},0\leq t\le T$$ ...
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39 views

Orthonormality of the columns of a matrix

I am studying orthogonal columns and matrices right now and I have encountered the following theorem: Theorem An $m \times n$ matrix $U$ has orthonormal columns if and only if $U^T U = 1$. Is it ...
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25 views

Condition number of positive definite matrix after rectangular orthogonal transformation on both sides

What is a lower bound on the condition number of $B A B^{T}$ (besides the trivial $\operatorname{cond}(B A B^{T}) \ge 1$) where $A$ is an $n \times n$ symmetric positive definite matrix, $B$ is a ...
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2answers
58 views

Normalizing a basis

Let the basis $B = \{1,x,x^2\}$ which is orthogonal. Now, I've seen the following: $$\|1\| = \sqrt {\langle 1,1\rangle} = \sqrt {4\cdot 1\cdot 1} = 2 $$ $$\|x\| = \sqrt {\langle x,x\rangle} = ...
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43 views

$\{x \mapsto e^{2\pi i k x} \mid k \in \mathbb{N}\}$ is orthonormal basis of $L^2$

I want to show that $\{x \mapsto e^{2\pi i k x} \mid k \in \mathbb{N}\}$ is orthonormal basis of $L^2((0,1); \mathbb{C}) =: X$. Of course the only problem is to show completeness. In our lecture we ...
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1answer
93 views

Proving that an orthonormal system close to a basis is also a basis

Let $\mathcal{H}$ be a Hilbert space and $(e_n)_{n \in \mathbb{N}} \subseteq\mathcal{H}$ be an orthonormal basis and $f_n$ be an orthonormal system such that $(f_n)_{n \in \mathbb{N}} ...
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1answer
14 views

Equivalent definitions of an orthonormal function

I want to prove that the following two definitions for an orthonormal function $\phi$, in terms of $kT$ time shifts, are equivalent. So let $T$ the symbol period and $k$ an integer. Definition 1 ...
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54 views

Orthonormal Basis of $L^2$

Theorem: ' ' The Orthonormal family $e_n(x)=e^{2\pi i n x},\ n\in\mathbb{N}$ is a basis for $\mathcal{L}^2([0,1])$.`` In this case, $\{e_n(x)\}_{n\in\mathbb{N}}$ being a basis would mean that any ...
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56 views

Orthonormal Basis of a function

An Orthonormal Family $\{e_k\}_{k\in\mathbb{N}}$ is a basis if and only if $$f=\sum^\infty_{n=1}\hat{f}(n)e_n \ \ \ \text{in} \ \mathcal{L}^2(\mathbb{R})$$ where $f\in\mathcal{L}^2(\mathbb{R})$ ...
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1answer
30 views

Why an orthonormal polynomial set over a continuous domain is not over a discrete one?

I would like to read the proof showing that a orthonormal polynomial set over a continuous domain is neither orthonormal nor complete over a discrete values on that domain. For example, Zernike ...
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4answers
79 views

Why are there infinitely many orthonormal vectors?

By Graham Schmidt process we can create infinitely many orthonormal vectors, but my doubt is that why is it not bounded by the dimensionality of the space ? Intuitively (geometrically) how can we ...
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1answer
23 views

Show that if $(e_n)$ is an orthonormal set in a Hilbert space $H$, the set of all vectors of the form $x=\sum c_ne_n$ is a subspace of $H$.

Show that if $(e_n)$ is an orthonormal set in a Hilbert space $H$, the set of all vectors of the form $x=\sum c_ne_n$ is a subspace of $H$. Hint: Take a Cauchy sequences $(x_r)$, where $x_r=\sum ...
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31 views

Expansion coefficients of an orthonormal basis must satisfy $c_n n^{1/2}\to 0$ as $n\to \infty$ for $\sum|c_n|^2$ to converge.

Expansion coefficients with respect to an orthonormal basis must satisfy $c_n n^{1/2}\to 0$ as $n\to \infty$ in order that $\sum|c_n|^2$ may converge. Is this true or false? Give a proof or ...
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1answer
28 views

A unitary space, interpreted as a Euclidean space

Let $(V, \gamma)$ be a $n$-dimensional unitary space. Let $V_{\mathbb{R}}$ be the vector space $V$, interpreted as a $2n$-dimensional $\mathbb{R}$-vector space. I first want to show that ...
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22 views

Using the definition of unitary / orthogonal operators explicity for matrices:

If A is unitary, then $$AA^* = A^*A = I, and\ A^* = A^{-1}$$ I want to see this explicitly for a very simple unitary matrix, say, take the column vector A = (1,0,0) and we regard this as a 3x1 ...
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40 views

Help on understanding this process of getting orthonormal basis

This method was taught by someone and I am trying to understand the process. Assume that we have $(1,2,3)$ as first vector. Since we need another vector that will make the dot product $= 0$ with the ...
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Bipolar toroidal coordinates - position vector, velocity and acceleration

Bipolar toroidal coordinates: $x = a \frac{\sinh\tau \cos\phi}{\cosh\tau-\cos\sigma}$ $y = a \frac{\sinh\tau \sin\phi}{\cosh\tau-\cos\sigma}$ $z=a \frac{\sin\sigma}{\cosh\tau-\cos\sigma}$ Would ...
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1answer
22 views

A linear operator T is normal if and only if there $[T]_\beta$ is normal, where $\beta$ is an orthonormal basis.

I am studying for a final exam and came across a sentence in my linear algebra textbook stating that "a linear operator T is normal if and only if there $[T]_\beta$ is normal, where $\beta$ is an ...
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1answer
30 views

Linear map, inner products and orthonormal bases

If $L: \mathbb{V} \to \mathbb{V}$ is any map and we have two inner products defined on $\mathbb{V}$, $[ , ]$ and $⟨ , ⟩$, and we pick two orthonormal bases w.r.t. each of these inner products, how can ...
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19 views

Linear map and inner products

How can one check that a map $L: \mathbb{V} \to \mathbb{V}$ preserves an inner product, provided that the basis with respect to the inner product is orthonormal?
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13 views

Orthonormal with respect to an inner product

What does it mean for a basis to be orthonormal with respect to an inner product?
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99 views

How many orthogonal matrices are there

this might sound like a stupid question, but what I mean is: You need $n \times n$ elements to define a square matrix $\in R^{n \times n}$. How many element do I need to define an orthogonal matrix? I ...
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4answers
48 views

Prove that $\|v \|^2= |\langle v, e_1 \rangle |^2 + \cdots + | \langle v, e_m\rangle |^2$

Suppose $(e_1,\cdots, e_m)$ is an orthonormal basis in $V$. Let $v \in V$ . Prove that $\|v\|^2= |\langle v, e_1 \rangle |^2 + \cdots + | \langle v, e_m\rangle |^2$ Let $v\in V$ and ...
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3answers
22 views

Completing an orthonormal basis of a plane to a basis for $\mathbb{R}^3$

I was asked to find an orthonormal basis for the plane $x + 2y +3z =0$. I found a regular basis, $(-2,1,0),(-3,0,1)$, and then performed the Gram-Schmidt process to find 2 orthonormal vectors that ...
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1answer
35 views

Is there anyway to check I have an an orthogonal and/or orthonormal basis?

I'm reading about Gram-Schmidt procedure in 3 dimensions. From what I understand the idea is to "fix" one of the vectors and alter the other 2 so they are all perpendicular. So say i have three ...
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1answer
31 views

Prob. 4, Sec. 3.4 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $(e_n)$ be an orthonormal sequence in an inner product space $X$. Then, for every $x \in X$, we have $$ \sum_{n=1}^\infty \vert \langle x, e_n \rangle \vert^2 \ \leq \ \Vert x \Vert^2.$$ Now ...
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1answer
42 views

Prob. 3, Sec. 3.4 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: How to derive the Schwarz inequality?

Let $\left( e_n \right)$ be an orthonormal sequence in an inner product space $X$. Then for every $x \in X$, we have $$ \sum_{n=1}^\infty \left\vert \langle x, e_n \rangle \right\vert^2 \ \leq \ ...
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28 views

Extend $(\frac{1}{2}, \frac{i}{2} ,\frac{-1}{2},\frac{-i}{2} )$ to an orthonormal basis for $\mathbb{C}^4$.

Consider $\mathbb{C}^4$ with the standard inner-product$ < , >$. Extend $(\frac{1}{2}, \frac{i}{2} ,\frac{-1}{2},\frac{-i}{2} )$ to an orthonormal basis for $\mathbb{C}^4$. How is this possible ...
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0answers
21 views

Maximally distant orthogonal matrices

I would like to construct a set of $k$ orthogonal matrices in $\mathbb{R}^{n \times n}$ with maximal summed pairwise distance (in terms of L2 operator norm). Any ideas? I am thinking of just doing ...
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33 views

Prove that a set is orthonormal on $L_2$

I would like to prove that the set of elements: \begin{equation} A_n(t)=\left\{\frac{1}{\sqrt{2\pi}}e^{int}\right\}_{n=-\infty}^{\infty} \end{equation} is an infinite orthonormal set, on space ...
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0answers
13 views

Show that an ONS $\{\phi_n\}$ is complete in $L^2(I)$ iff $\sum_{n=1}^\infty(\int_{[a,x]}\phi_n)^2=x-a$ for all $x\in I$.

Suppose $I=[a,b]$ is an interval of the line. Show that an ONS $\{\phi_n\}$ is complete in $L^2(I)$ iff $\sum_{n=1}^\infty(\int_{[a,x]}\phi_n)^2=x-a$ for all $x\in I$. My Work: If we suppose ...
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25 views

Planes And Lines

Given :Point $A(1,2,4)$ and plane $P: x-y+z+2=0$ How to find coordinates of point $A'$ the symmetric of point $A$ with respect to plane $P$
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Question about projecting vector onto Subspace

Consider subspace $S \in R^4$, spanned by the vectors: $v_1 = (1,0,-1,1)^T$ and $v_2 = (1,1,1,1)^T$. Let $v = (1,-1,-1,3)^T$. I want to find the projection of $v$ onto $S$; that is, find the ...
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42 views

Determining if this is an orthonormal set.

$p(x)\:=\:a+ax+ax^2$ and $q(x)\:=\:b+bx+bx^2$ are vectors in $P_2$ The inner product is the dot product: $$\langle p,q\rangle=a_0b_0+a_1b_1+a_2b_2$$ the set is: ...
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1answer
28 views

Is there some clever way of finding out this SVD?

I need to find the singular value decomposition of $$\pmatrix{a&-b&0\\b&a&-b\\0&b&a}$$ I already determined sigma to be: $$\Sigma = ...
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1answer
22 views

Fundamental Subspaces: Orthonormal Bases

In matlab, we are asked to set A=rand(5,2)*rand(2,5) then to set Q=orth(A), w=null(A'), S=[Q W] the matrix S should be orthogonal. Why? (I have no clue on how to answer this)
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1answer
19 views

Why does $A=UDU^H$ = $\lambda_1 u_1 u_1^H + \lambda_2 u_2 u_2^H$?

Just trying to figure out why this is true: $$A=UDU^H \quad\Leftrightarrow\quad A= \lambda_1u_1u_1^H + \lambda_2u_2u_2^H$$ $U$ is a unitary matrix composed of the eigenvectors of hermitian matrix A. ...
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1answer
55 views

Orthonormal and/or Orthogonal Basis of a Pair of Vectors

I was hoping someone could verify if this is the correct way to answer this problem: Let $\mathbb{R^{2}}$ have the standard dot product. Classify the following pair of vectors as (i) basis, (ii) ...
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21 views

Why is $||\vec{x}^\perp||^2=0 \,\forall\, \vec{x}\in \mathbb{R}^n$?

Consider the decomposition of $\vec{x} \in \mathbb{R}^n$ where $\vec{x}=\text{proj}_V\vec{x}+\vec{x}^\perp$. From Pythagoras, $||\vec{x}||^2=||\text{proj}_V\vec{x}||^2+||\vec{x}^\perp||^2$. It ...
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3answers
47 views

Gram-Schmidt Process and Orthogonal Components

Let the Gram-Schmidt process transform the vector system $(a_{1}, ..., a_{n})$ into the system $(b_{1}, ..., b_{n})$. Show that the vector $b_{k}$ is the orthogonal component of the vector $a_{k}$ ...
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2answers
60 views

Find orthonormal basis of $\mathbb{R}^3$ with a given span of two basis vectors

What is an orthonormal basis of $\mathbb{R}^3$ such that $\text{span }(\vec{u_1},\vec{u_2})=\left\{\begin{bmatrix}1\\2\\3\end{bmatrix},\begin{bmatrix}1\\1\\-1\end{bmatrix}\right\}$? I was thinking I ...
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1answer
56 views

How does equality in Bessel's Inequality prove an orthonormal complete sequence?

I've been searching around for an answer to this question on the web for some time, but I keep coming up short (it may very well be that I don't have the right terms to be searching with). In any ...
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4answers
98 views

Eigenvalues and Eigenspaces of a Projection

Let $P$ be the orthogonal projection onto a subspace $E \subset V$ ($V$ being an inner product space) with $\mathrm{dim(V)}=n$, $\mathrm{dim(E)}=r$. Obtain the eigenvalues and eigenspaces, along with ...
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1answer
17 views

Proving orthonormality of system by sum of fourier coefficients

Let $f\in L^2(\mathbb R)$. Prove the system $\{f(t-n)\}_{n\in\mathbb{Z}}$ is orthonormal if and only if $$\sum_{k\in\mathbb{Z}}|\hat{f}(\omega+2\pi k)|^2\equiv 1$$ I have no clue how to prove ...