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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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Why do we need projection in the definition of the Stokes operator?

$\DeclareMathOperator{\div}{div}$ $\def\bu{\mathbf{u}}$ Let $D$ be the square $[0,1]^2$ and consider the following space: $$ V:=\{\bu: \bu\in H^2(D)^2, \div \bu=0, u|_{\partial D}=0 \}. $$ Introduce ...
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Problems with determining the orthonormal basis regarding an inner product $B$

Can somebody tell me where I made a mistake? How would you approach such an exercise? Was my way too complicated? $ B:\mathbb{R}^3\times \mathbb{R}^3\to \mathbb{R},\quad B(x,y)=\sum \limits_{i\neq j}^...
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Diagonalizing a real normal matrix

Given the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{pmatrix}$, how would I find a real orthogonal matrix $P$ such that $PAP^t$ is a diagonal matrix? ...
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Integrate orthogonal function over solid angle

How do I integrate a product of Legendre polynomials over a volume? So I understand that bunch of complete basis orthogonal basis as well. i.e. for Legendre polynomial, $\int_{-1}^1P_n(x)P_m(x)dx=\...
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Why is the maximum value of an orthonormal dictionary of size $N \times N$ less than that of another having size $M \times M$, where M>N?

I was trying to compare the maximum value in DCT dictionary (representing an orthonormal dictionary) of size $N \times N$ to that of another DCT dictionary of size $M \times M$, where $M>N$. I ...
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Relation between norm of a vector and sublattice to which it belongs.

Given four rational numbers $m, n, p$, and $q$, define the lattices $L_1 = \{ (m\cdot x, n\cdot x) : x \in \mathbb{Z} \}$ and $L_2 = \{ (p\cdot y, q\cdot y) : y \in \mathbb{Z} \}$. That is, the ...
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Orthonormal basis to a four dimensional space with dot product well defined

Suppose I have 4-dimensional space with a dot product well defined. I have two vectors that are orthonormal, how can i find two more vectors to form a complete basis to this 4-dimensional space? I ...
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Does always a $B$ with orthonormal rows/column be found so that $BP=0$?

I am given, $P_{n\times n}$ column stochastic, I need to see whether a matrix of suitable order $B$ with an orthonormal column or rows can be constructed so that $BP=0$? I started trialing like this: ...
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Approximating orthonormal basis in Hilbert space by orthonormal basis from dense subset

Consider a Hilbert space $H_2$ with norm $\|\cdot\|_2$. Consider a linear space $H_1 \subset H_2$ so that $H_1$ is dense in $H_2$. Consider an orthonormal sequence $(h_k)_{k \in \mathbb{N}}$ in $H_2$. ...
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Orthonormal $\{e_n\}$ is complete in $L^2[a,b]$ iff $\sum_{n=1}^{\infty} \mid \int_{a}^{x} e_n(t) dt \mid^2 = x-a$ for all $x \in [a,b]$.

For $n \in \mathbb{N}$, let $\mathcal{E}$=$\{e_n\}$ be an orthonormal sequence in $L^2[a,b]$. We need to show that $\{e_n\}$ is complete in $L^2[a,b]$ iff $\sum_{n=1}^{\infty} \mid \int_{a}^{x} e_n(t) ...
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Orthonormal Basis In $l_1$ Norm

If I am working on finite dimensional innerproduct space $\mathbb R^m$, and $\{v_{1},\dots,v_{m}\}$ be a orthonormal basis for $\mathbb R^m$, can I say that they are orthonormal in the sense of $l_1$ ...
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Determine orientation of standard orthonormal bases

Let $\vec{e_1},\vec{e_2},\vec{e_3}$ be the standard orthonormal base $(1,0,0),(0,1,0),(0,0,1)$ which is positively oriented. Determine the orientation of each of the following bases: $\vec{e_1},\vec{...
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Gram Schmidt Process for a Complex Vector Space

Suppose I have certain independent vectors, say $\lvert V_1\rangle$ and $\lvert V_2\rangle$, which span a 2-dimensional subspace of a given Complex Vector Space on which inner product is defined, how ...
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Separable Hilbert Space

Let $\mathcal{H}$ be a separable Hilbert space with complete orthonormal system $e_{1},e_{2},\ldots$ (a) Let $T:\mathcal{H}\rightarrow\mathcal{H}$ be a continuous linear operator. Suppose that there ...
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Sum of Fourier coefficients less than the norm of the respective vector

I have the following question to complete. Let $X$ be an inner product space. Let $(e_{j})_{j\geq1}$ be an orthonormal sequence in $X$. Show that, \begin{align} \sum_{j=1}^{\infty}|(x|e_{j})(y|e_{j})|...
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Calculate an orthonormal basis

We have the following Gram-Schmidt algorithm: I want to calculate for the following vectors an orthonormal basis, with an accuracy of $\epsilon=5\cdot 10^{-3}$. \begin{equation*}a_1=\begin{pmatrix}...
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1answer
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Explicit formula for inner product given orthonormal basis

Suppose $V$ is a $\mathbb C$-vector space and $v_1, v_2, \ldots v_n \in V$ (given explicitly, for instance as vectors with given elements) form its orthonormal basis relative to some inner product $\...
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Norm of Fourier series

I am reading the proof of the statement that no non-zero multiplication operator on $L^2([0,1])$ is compact in this post. And I would like to address it as a seperate post as I am only curious about ...
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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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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
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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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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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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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196 views

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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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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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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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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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
194 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
27 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 ...