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Questions tagged [orthonormal]

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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orthonormal basis of infinite dimensional Hilbert space H is not a basis of H as vector space?

Apparently the orthonormal basis $(e_n)_{n\in \mathbb{N}}$ of the Hilbert space $H$ (in special case, infinitly dimensional) is not a basis of $H$ as a vectorspace. Is there a way to prove this?
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Surface integrals, positive or negative normal?

I'm unsure how to decide whether the normal should be positive or negative in $\hat{n}dS=\pm h_2 h_3 e_1 du_2 du_3$, where $h_i$ are the scale factors, $e_i $ are the base vectors, and $u_i$ are the ...
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If $\mathbf{av}_1 = \mathbf{av}_2 = 0$ then does orthonommalising $\mathbf{v}_1, \mathbf{v}_2$ change this?

Suppose $\mathbf{a^\top, v_1, v_2}$ are $2 \times 1$ vectors. By applying the Gram-Schmidt process and orthonommalising so that we now have $\mathbf{e_1, e_2},$ is: $$a\mathbf{e}_1 = a\mathbf{e}_2 = ...
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Orthogonal projection in Hilbert space

Projector $P\neq0$ is orthogonal projection if and only if satisfied some of the following mutually equivalent conditions: a) $P$ is self-adjoint, $P=P^*$ b) $P$ is normal, i.e. $P^*P=PP^*$ c) $P$ ...
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Orthogonal matrices with eigenvalues equally spaced over unit circle?

We know that the eigenvalues of orthogonal matrices have norm 1, and thus they are all on the unit circle. However, I wonder if there is a way to construct a orthogonal matrix (with real entries) ...
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“minimization” problem in Euclidean space related to orthonormal basis

I have stumbled upon an exercice for second year undegraduate student majoring in economics which I find quite demanding. I have an idea for the solution, but it seems awfully complicated, and I am ...
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Convergence of sum $f_n(x)=\sum_{l,k} w_{l,n} w_{l,k} x^k$ , with $w$ expansion coefs of an orthonormal system

Good day, Let $\{P_k\}$ be a complete orthonormal system (Fourier series, Legendre-Fourier series, etc..) on interval $(a,b)$ which can be expanded into powers : $$ P_n = \sum_{k=0}^\infty w_{n,k}x^...
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1answer
28 views

A complete orthonormal system $\{e_i\}^\infty_{i=1}$ in $H$ is a basis in $H$

I'm studying about Hilbert space from a book of functional analysis and I just read this theorem (2.1.10) and its' proof. I cannot understand why $(y-x)\perp e_i$? why is it implied?
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Criteria to find a common non orthonormal basis for two linear operators

I can't find any criteria to determine, in finite dimension, if two operators has a non orthogonal common basis, for example, given two operators A and B if I check $A=A^+$ $B=B^+$ $[A, B] =0$ In ...
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Do orthonormal matrices have a geometric interpretation or contextual importance? [closed]

I understand how certain matrices can be orthonormal and the conditions necessary for that. I don't understand its geometric relevance. For instance in a vector space the axis would be orthonormal and ...
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How can we decompose the identity matrix given a set of orthonormal vectors?

Let $A$ be a positive semidefinite (P.S.D) matrix with distinct set of eigenvalues. since it is P.S.D its eigendecomposition is as follows for eigenpairs of $(\lambda_i,v_i)$ $$ A= \begin{bmatrix} ...
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Applying sine orthogonality

I have a confusion as to why this is a viable procedure in the following image: $$ \sum_{n} B_n \int_{0}^{a} \sin \left( \frac{n \pi x}{a}\right) \sin \left( \frac{m \pi x}{a}\right) = \sum_{n} B_n \...
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1answer
52 views

Non total orthonormal set in a non Hilbert inner product space

Suppose there exist a subset $M$ of an inner product space $X$, and the orthogonal complement of $M $ is the zero vector. If $X $ is a Hilbert Space then the span of $M $ will be dense in $X $, but ...
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Creating an orthonormal basis with Gram schmidt procedure error.

I have a question which says the following: Let $V$ be the span of $v_{1}=(0,1,2)$, $v_{2}=(-1,0,1)$ and $v_{3}=(-1,1,3)$. Construct an orthonormal basis $B'$ for $V$ (usual dot product). I ...
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Finding orthonormal basis of subspace?

Question I am completely lost on this problem. I know how to find it using Gram-Schmidt but I'm unsure of how to even find the subspace in this case, or how I would graph any of this. Is there another ...
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22 views

How to determine if three vectors form a basis for a subspace?

This is a follow up question ( math.stackexchange.com/q/3018473); i'm interested in understanding some other part of the problem. I have three vectors, v1, v2, v4, which are linearly independent. ...
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Does this specific SO(4) matrix have to be block-diagonal?

So I have a specific real $4\times4$ matrix $\mathbf{P}$ given by \begin{align} \mathbf{P}= \begin{pmatrix} p_{11} & -p_{21} & p_{13} &-p_{23}\\ p_{21} & p_{11} & p_{23} & p_{...
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Finding Outer Normal of Supporting Hyperplane

Let $\mathcal{M}:=\{x \in \mathbb R^{2}: x_{2} \geq |x_{1}| \}$. Find all outer normals $y \in \mathbb R^{2}$ of supporting hyperplanes to $\mathcal{M}$. My ideas: Let supporting hyperplane $\mathcal{...
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For each value of $t$, find an orthogonal basis of the span of the vectors:

$u_1 = (1,t,t)$, $u_2 = (2t,t+1,2t-1)$, $u_3 = (2-2t,t-1,1)$ Any help would be appreciated, if you could explain how to work such questions out
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Projection on a subspace

An inner product is defined on $P_3$ such that $<f, g>$ $=$ $\int _{-1}^1\:f\left(t\right)g\left(t\right)dt$. What is the orthogonal projection of $p(x)$ $=$ $x^3$ onto the subspace $S$ $=$ $\...
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Orthogonal Projection on a Polynomial Space

An inner product is defined on $P3$ such that $<f, g>$ $=$ $\int _{-1}^1\:f\left(t\right)g\left(t\right)dt$. What is the orthogonal projection of $p(x)$ $=$ $x^3$ onto $P2$? So I got that $f_1\...
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If we have an orthonormal basis of $\mathbb{R}^n$ how we can describe every vector in $\mathbb{R}nT$ using them?

Let $S = \mathbb{R}^n$ be a subspace with dimension $n$. Also, let $\{\phi_j\}_{j=1}^k$ be $k$ orthonormal vector that describes another subspace $\Psi_k \subseteq \mathbb{R}^n$ with dimension $k$ ...
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43 views

Find out if a operator is self-adjoint or orthogonal

Let $V$ be a $\mathbb{R}$ inner product space, and $B=\left \{v_1, v_2, v_3 \right \}$ basis of $V$, with $||v_i||=1$ $\forall i=1,2,3$, and $<v_1, v_2>=<v_1, v_3>=0$ and $<v_2, v_3>=...
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42 views

Why are the Fourier Series an orthogonal basis?

The Fourier Series of a function $y(x)$ is its expansion into sines and cosines: $$y(x)= a_0+a_1\cos(x) +b_1\sin(x)+a_2\cos(2x)+b_2\sin(2x)+...$$ An Orthogonal Basis for an inner product space $V$ ...
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1answer
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Showing that a normal matrix with real eigenvalues is Hermitian. A question about properties orthonormal matrices.

This problem is part of exercise 2.17 in Nielsen and Chuang's textbook, and has been already answered on this site in this post. I understand that because $A$ is normal, it can be orthogonally ...
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224 views

How do I find a tangent plane without a specified point?

I was having a problem finding the points on $z=3x^2 - 4y^2$ where vector $n=<3,2,2>$ is normal to the tangent plane. How do we calculate the tangent plane equation without a specific point to ...
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Prove that if $u$ is a least squares solution of $Bx = b$ then $u$ $=$ $R^{-1}$ ($Q^T$) $b$.

Let $B$ be an $m \times n$ matrix, whose columns are linearly independent. Suppose that $B$ has a $QR$ factorization, i.e., $B = QR$ where $Q$ is an $m \times n$ matrix with orthonormal columns and $R$...
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I know symmetric matrix $S = QDQ^T$, but how can matrices with form ADA be symmetric?

I have learned that a symmetric matrix must be able to be written in form of $S=QDQ^T$ where Q is the orthonormal eigenvectors. But I saw an example that display a symmetric matrix in the form of $S = ...
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1answer
67 views

Sum of rank-one matrices equals identity

Let $v_1,\dots,v_n\in \mathbb{C}^n$ be vectors satisfying $$ v_1v_1^* + \dots + v_n v_n^* = I $$where $I$ is the identity matrix and $v^*$ denotes the conjugate transpose. These vectors are clearly ...
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How to manage distance between points in an orthonormal base

I am currently working on a java application trying to solve as efficiently as possible the travelling salesman problem with the various proven methods or ones which I found relevant. The ultimate ...
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1answer
80 views

Show that a symmetric and idempotent matrix $P$ is the projection matrix onto some subspace.

I am reading "Seminar of Linear Algebra" by Kenichi Kanatani. In this book, there is the following problem: Show that a symmetric and idempotent matrix $P$ is the orthogonal projection matrix onto ...
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1answer
26 views

Rotate vector orthonormally into a target vector

I have $k$ vectors $v_i\in\mathbb{R}^n$, mutually orthogonal. I would now like to rotate them in the $k$-dimensional subspace spanned by the $v_i$ such that $v_0$ ends up at the given target vector $w\...
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Align normal to plane

How do I align normals to a plane. In this specific case I have a bunch of points and normals corresponding normals. I am projecting the points onto the yz plane and need to know what the normals ...
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1answer
29 views

How to expand function with summation and equation inside the L2-norm?

I want expand an L2-norm with some matrix operation inside. Assume I have f(c) = $\sum_{i=0}^n ||x_i - c||_2^2 $ Should I do: $\sum_{i=0}^n [ ||x_i||_2^2 - ||c||_2^2 ] $ $\sum_{i=0}^n [ ||x_i||_2^...
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How do I find the normal of the equation $x^4 + 5y^4$? at $(2,1)$

I know that the normal of a line is a line with a slope that is orthogonal to the line at a given point. The equation in question is $x^4 +5y^4 = 21$ In order to calculate the tangent line of the ...
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1answer
27 views

orthonormal set depending on variable

Let $A(t) = \{x_1(t), x_2(t),..., x_n(t) \}$ with $0 \leq t \leq 1$ where $x_i(t) \in \mathbb{R}^n, \forall i$. I would like to construct a set $A(t)$ such that $A(t) $ is an orthonormal set, i.e. $...
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97 views

Orthonormal Eigenbasis of the reflection matrix

So this question is in a way more about computation than theory, because I feel pretty confident in the latter but yet can't get the former to work. What I seek is given $\begin{pmatrix} \cos(\...
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orthogonalization and orthonormalization

I have to construct a diagonalizable and orthogonal matrix starting from this quadratic form $Q(x_1,x_2,x_3)=-2x_1x_3+2x_1x_3-2x_2x_3$ in order to reduce it in canonic form with a variables change. I'...
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1answer
47 views

Orthogonal Basis and orthogonal projection

I have this problem: Let $V = \{(x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1 + 3x_2 -5x_3 - x_4 = 0\}$ Find an orthogonal basis for $V$. What's the closest point to the origin over the plane $x_1 + ...
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If $v = \sum_{i} \xi_{i}v_{i}$ then $\sum \xi_{i}^2 = 1$?

If $v$ is a unit vector and $v = \sum_{i} \xi_{i}v_{i}$ where $v_{i}$ are orthonormal vectors. Then how do we prove $\sum \xi_{i}^2 = 1$ ? I thought that we have $<v,v_{i}> = \xi_{i}$ by ...
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An orthonormal base of a set of waveforms

I'm trying to resolve an excercise of telecomunication that ask me to crate an ortonrmale base for this set of waveform. $\ s_{1,2,3,4} (t) = \pm Acos(2 \pi f_0t+\varphi_0) \pm Asin(2 \pi f_0t+\...
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3answers
60 views

How to compute coordinates of three points in the standard basis?

I have the orthonormal basis for $b_1 = (-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$, $b_2 =(-\frac{\sqrt{2}}{2}), (-\frac{\sqrt{2}}{2})$ for $\mathbb{R}^2$. I need to compute the coordinates of the ...
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How do I prove orthogonality if the lengths of two vectors are equal?

If $||v|| = ||w||$, how can we show $(v + w)$ and $(v - w)$ are orthogonal? I can't find a way to show $(v + w) * (v + (- w)) = 0$
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Hilbert space and orthonormal basis.

Let $H$ be a Hilbert space and let ${e_n} ,\ n=1,2,3,\ldots$ be an orthonormal basis of $H$. Suppose $T$ is a bounded linear oprator on $H$. Then which of the following can not be true? $$(a)\quad T(...
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1answer
35 views

Functional analysis orthogonal sequence. [duplicate]

Let $\{{e_n}\}\ n=1,2,3,...$be an orthonormal sequence in a Hilbert space $H$ and let $x\not=0 \in H$ then $<x, e_n> \to 0$ as $\ n \to \infty$. By Bessel's inequality we have $\sum_{n=1}^{\...
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3answers
212 views

Forming an orthonormal vector when you already have two perpendicular vectors

So I understand the requirements for an orthonormal basis and everything around it. However, there's one thing I am missing: Suppose you have two vectors which are orthonormal $u_1$ and $u_2$. ...
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1answer
37 views

Does $P$ have to be orthonormal in orthogonal decomposition of a symmetric matrix $A=PDP^T$?

I want to diagonalize the following matrix: $$A=\begin{bmatrix}1 & -4 \\ -4 & -5\end{bmatrix}$$ I computed its eigenvalues, $\lambda_1=3$ and $\lambda_2=-7$, and got that the eigenspace ...
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2answers
49 views

Find eigenvalues/vectors of $A=\mathbf{u}\mathbf{v}^T+\mathbf{v}\mathbf{u}^T$ where $\mathbf{u},\mathbf{v}$ are orthonormal

Our problem is to compute the eigenvalues and eigenvectors of two matrices formed by products of orthonormal vectors, $\mathbf{u}$ and $\mathbf{v}$: $A=\mathbf{u}\mathbf{v}^T+\mathbf{v}\mathbf{u}^T$ ...
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3answers
49 views

Calculating an orthonormal basis of $\{(x,y,z) \mid 2x + y -z = 0\}$

We have the real euclidean vector space $\mathbb{R}^3$ with the standard inner product and the standard basis $B = (e_1, e_2, e_3)$. $W \subset \mathbb{R}^3$ is the subspace which is defined by: ...
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49 views

Optimal orthonormal basis in $\mathbb{R}^p$

Assume $\{c_i\}_{i=1}^p$ is an orthonormal basis of $\mathbb{R}^p$. $\{\alpha_i\}_{i=1}^p$ and $\{\beta_i\}_{i=1}^p$ are two sets of positive real numbers such that $\alpha_1>\alpha_2>...>\...