Questions tagged [orthogonality]

This tag can be used to refer to the orthogonality of a set of vectors, a matrix or a linear transformation.

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Where I went wrong in this chain of arguments - Linear algebra

Let $AB = I_n$. $A$ and $B$ are nonsingular, square matrices of size $n$. Let $A_{r1}$ be the first row of $A$. The products $A_{r1} B_j = 0, j \in \{2,\dots,N\}$. $B_j$ is the $j^{th}$ column of $B$. ...
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How to prove the form of the functions which are symmetric with respect set?

Let $G$ be any set of orthogonal linear transformations of $R^{n}$ onto itself. A function $f$ is said to be symmetric with respect to $G$ if $$ f(A x)=f(x), \quad \forall x, \quad \forall A \in G . $$...
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Generate a uniformly sampled orthonormal matrix that 'rotates' $k$ vectors $x_0 \in \mathcal{R}^{n \times k}$ into $y_0 \in \mathcal{R}^{n \times k}$

We know that orthonormal matrices $H \in \mathcal{R}^{n \times n} $ are rotation matrices. Is there a general method to uniformly generate rotation matrices that can rotate a given set of vector $x_0 ...
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Do we lose orthogonality of Bessel functions when we change interval

This is the basic definition of integral when you calculate integral product of orthogonal Bessel functions. What happens when you change integral bounds from [0,a] to [b,c]? Do you lose the ...
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The Numerical Issues of Computing The QR-Decomposition Using CholeskyQR

I have an application where I need to compute the QR-decomposition (in fact I only need $R$) of a matrix $A\in \mathbb{R}^{m \times n}$ ($m > n$). The matrix has a fixed number of columns, but the ...
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The norm and the Fourier coefficient if the inner product $\langle f(x), f(x)\rangle$ is negative.

For a real function $f_m(x)$ orthogonal with respect to $w(x)$ where $w(x)>0$. We have inner product $$ \langle f_m,f_n\rangle_{w}=\int_{x_0}^{x_0 + T}{w(x)f_m(x)f_n(x)dx} $$ The norm is $||f_m||_{...
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Proving a formula about inner product of a vector and its projection onto a orthogonal complement of the column space

We have $h_1,...,h_{i},...,h_K \in \mathbb{C}^N, N\gt K$ and they are linearly independent. Then we define $v \triangleq \prod^{\bot}_{[h1,...,h_{i-1}\ ,\ h_{i+1} \ ,...,h_K]}h_i $, where $\prod^{\bot}...
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orthogonal polynomials and determinant of jacobi matrix

In the book 'Orthogonal Polynomials of Several Variables' by Charles F. Dunkl and Yuan Xu page 11 is the picture below. I assume the matrix in the picture is related to Corollary 1.3.10 For the case ...
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Orthogonality of the Legendre polynomials with respect to $L_2$ (Integration by parts)

I want to show, that the Legendre polynonmials are orthogonal with respect to scalar product in $L_2[-1,1]$. The Legendre polynomials are defined as follows: $$P_n(x) = \left( \frac{2n+1}{2}\right)^{\...
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If $P_1$ and $P_2$ are orthogonal projectors, then $\mathrm{tr}(P_1 P_2) \leq \mathrm{rk}(P_1 P_2).$

Prove or provide a counterexample: If $P_1$ and $P_2$ are orthogonal projectors, then $\mathrm{tr}(P_1 P_2) \leq \mathrm{rk}(P_1 P_2),$ where $\mathrm{tr}$ denoted the trace and $\mathrm{rk}$ the ...
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Is the orthogonal complement projector a projection operator?

Let $X$ Hilbert space and $U \subset X$ convex closed. Then we can define the projection operator on $U$ $P \colon X \to X$ such that $P x \in U$ is the orthogonal projection of $x \in X$. Moreover $P$...
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Help with understanding the Gram-Schmidt Process

Let $U=\langle x_1,x_2,x_3\rangle \subseteq \mathbb{R^4},$ where $$x_1=\begin {pmatrix} 3\\4 \\0\\0 \end {pmatrix}, \ x_2=\begin {pmatrix} 1\\3 \\1\\1 \end {pmatrix},\ x_3=\begin {pmatrix} 0\\5 \\5\\7 ...
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In a product space $E$, and subspace $F \subset E$, $\|u-v\| \geq \|v\|, \forall v \in F \iff u \in F^\perp$

I'm having trouble proving that in a product space $(E, \langle \cdot, \cdot\rangle)$, and subspace $F \subset E$, $\|u-v\| \geq \|v\|, \forall v \in F \iff u \in F^\perp$. In fact, the reciprocal ...
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Is the Sy basis of Spin not orthogonal?

According to Wikipedia on Spin $1/2$ particles, the vectors of the $S_y$ basis, with respect to the $S_z$ basis, are: \begin{bmatrix} 0.707 \\ 0.707*i \end{bmatrix} \begin{bmatrix} 0.707 \\ -0.707*i \...
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Spherical harmonics orthogonality

I've been struggling with this integral $$ \int_0^{2\pi}\int_0^{\pi} \sin(\theta) e^{-i\phi} Y_{l m}(\theta,\phi) Y^*_{l'm'} (\theta,\phi) \mathrm d\theta \mathrm d\phi $$ I've tried to use the ...
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A vector perpendicular to two other vectors

Let $u,v \in {R}^{3}$ be linearly independent . Find a third vector in ${R}^{3}$ that is perpendicular to both ${v}^{\perp}$ and $u$ , where ${v}^{\perp}$ is the orthogonal projection from $v$ onto $u$...
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A segment having equal angles with two equal segments is perpendicular to the connecting line

Let $a,b,c \in \mathbb{R}^3$ be unit vectors, and suppose that the angles between $a,b$, and between $a,c$ are equal. Is there an elementary, geometric, computation-free proof that $a$ is ...
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Universal Properties of Orthogonal Matrices

I wanted to ask this question since I have seen conflicting viewpoints on it. Are orthogonal matrices necessarily symmetric? I do not believe so but some website said they were so I need to confirm. (...
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Orthogonal complement to symmetric functions in $L^2$

Given $$M=\{ f\in L^2([-1,1])\; | \; f(x)=f(-x) \text{ for almost every } x\in [0,1]\}$$ I have already proved that $M$ is a closed subspace of $L^2([-1,1])$, but have yet to find $M^\bot$. Obviously, ...
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Coordinate Vector Relative to Orthogonal Basis

I'm studying for my Linear Algebra final and have came across a problem that I cannot find any examples of in my book and very few online. I wish to know if my solution takes the right steps and ...
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Double orthogonal complement of a finite dimensional subspace w.r.t. a nondegenerate symmetric bilinear form on an infinite dimensional space

Let $U$ be a finite-dimensional subspace of an infinite-dimensional space $V$ equipped with a nondegenerate symmetric bilinear form $\phi$. Is it necessarily true that ${(U^{\perp})}^{\perp}=U$? If ...
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Matching Coefficients of Fourier Series with Separation of Variables Solution for Discontinuous Boundary Conditions: 2D Slab Conduction

I am trying to find the temperature profile in a 2D domain with steady heat conduction. The non-dimensional domain is shown below. Domain dimensions, coordinate system, boundary conditions, and ...
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Inner product space and orthonormality

We have an orthonormal sequence $\{e_n\}$ on the inner product space $V=C([0,\,1],\,\Bbb R)$. Let $h\in V$ and assume $\langle h, e_n\rangle=0$ for integers $n\ge0$. Show $\langle h, p\rangle=0$ for ...
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Orthogonal project of a vector $\vec{y}$ that sit in the same plane (aka span($u_1, u_2$)) such that $\vec{y},\vec{u_1}, \vec{u_2} \in \mathbb{R^3}$

Confirmed that $u_1,u_2$ are indeed orthogonal by taking their dot product which equals zero. Correct answer = $\begin{pmatrix}-1\\ -1\\ 6\end{pmatrix}$. Geographically, mapped out in geogebra: we see ...
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Orthogonal projection of $D=\begin{pmatrix}1 & 1\\ -3 & 1\end{pmatrix}$ on $W=\{A\in M_{2\times 2}(\mathbb{R})~|~Trace(A)=0\}$ in product space $V$.

This was an assigned homework problem (Turned in 4/14): $V=M_{2\times 2}(\mathbb{R})$ with the inner product $\langle A,B\rangle = Trace(B^t,A),~D=\begin{pmatrix}1 & 1\\ -3 & 1\end{pmatrix}$, ...
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Eigenvectors of an Orthogonal Matrix

It is true that for orthogonal matrices, all eigenvectors of a single eigenvalue are orthogonal to all other vectors of different eigenvalues(distinct eigenbases are orthogonal). My question is are ...
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Projection onto union of two affine subsets

Let $\ C_1=\{\ x\in {R^n} \ |\ \langle u,x\rangle\leq\alpha\ \}\ $ and $\ C_2=\{\ x\in {R^n} \ |\ \langle u,x\rangle\leq\beta\ \}. $ Give the orthogonal projection of $x\in{R^n}$ onto $\ C_1\cup C_2....
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Brownian Motions, Integrals, and Orthonormal Functions

I am out of my element with a topic I am working on. I think I have dwindled down the part I am stuck on to the following. If $\phi_n(t)$ is a sequence of orthonormal (eigenfunctions) functions and $...
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How to understand the relationship of the fundamental subspaces in these big pictures?

I am struggling like 6 hours to understand what this content in the middle mean. Can u get me some clue to interpret it? I understand all the stuff on the sides. So, row space is perp. to null space ...
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Are there radial basis functions $\phi_n(x)$ which can be factored as $f_n(x) x^{\alpha}$ for $0<\alpha<1$ and $n>0$?

I am looking for a set of radial basis functions $\phi_n(x)$ on $\mathbb{R}^+=[0;\infty[$ which satisfy some othogonality condition $$ \int x \phi_n(x) \phi_{n'}^*(x) dx =\delta_{nn'}$$ and "...
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Smoothness of a real smooth normed space yields the Birkhoff orgonality is additive on the right

In a normed space $X$, the Birkhoff orthogonality is defined as following: $$x\perp_B y \quad ‎ ‎\Leftrightarrow \quad \| x+\lambda y\| \geq \| x\|.$$ A well known characterization of the Birkhoff ...
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How to find eigenvectors of a matrix given only eigenvalues?

I'm trying to solve this problem: Given a $3\times 3$ matrix with eigen values $3,0,-1$, are their associated eigen vectors $(v_1,v_2,v_3)$ orthogonal to each other? My thought process is that we ...
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Finding a basis of a subspace which is perpendicular

Let $V = \text{span}\{(1, 1, 0, 1, 1), (1, 2, 0, 2, 1), (1, 3, 3, 3, 1)\}$ be a subspace of $\mathbb R^5$. I found using Gram-Schmidt process that the orthonormal basis of $V$ is $\{(0.5 , 0.5 , 0 , 0....
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Is there a formula for the norm of an orthogonal projection?

In all introductory linear algebra texts there is a discussion on orthogonal projection. Let $u = w_1 + w_2$, where $w_1$ is the projection of $u$ along $v$ and $w_2$ is projection of orthogonal to $v$...
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It is known that the line doesn’t pass through the origin, what is projection of vector $\vec{a}$ onto line L

Question : Let $L$ have a vector from $\vec{x}$ = t $\begin{bmatrix} -1 \\ 1\end{bmatrix}$ + $\begin{bmatrix} 0 \\ 1\end{bmatrix}$, and let $\vec{a}$ = $\begin{bmatrix} 2 \\ 6\end{bmatrix}$...
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Gram Schmidt process against orthogonal basis W

Another question that has a wrong answer from people adopting it. Am I wrong or the textbook wrong? Answer from book: My ans using the Gram-Schmidt process: such that $\vec{x_1} and \vec{x_2}$ are ...
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Understanding orthogonal projection of $\vec{y}$ on to span of orthogonal set, with an example

This is to verify if there's an issue with my understanding or if there's issue with the textbook. There seem to be also a previous question here on exactly the same, hoping to help myself and future ...
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Approximate orthogonality between two sets of Hermite functions.

Consider the set of Hermite functions $\{\phi_{n}(x,\varepsilon_{1})\}_{n}:= A$ defined below. \begin{equation} \label{eqn:funcs} \phi_{n}(x,\varepsilon_{1}) = \frac{\sqrt[8]{1+\big(\frac{2\...
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How to compare orthogonal vectors?

I doubt if I am asking this question correctly but for what it’s worth I have a set of orthogonal vectors for which I would like to pick from another set the closest orthogonal vector from it to my ...
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Describe an orthogonal subspace of prehilbert space

I'm working on pre-Hilbert spaces and there is something that I don't know how to prove in an exercise: In the pre-Hilbert space $\mathcal{C}([-1,1])$, under the inner product $(f|g)=\displaystyle{\...
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On orthogonal and non-orthogonal coordinate basis conversion

How exactly are orthogonal and non-orthogonal coordinate bases describes exactly? I'm a beginner in the field and hope to understand them better.
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True or false: Every set of orthogonal vectors in $\mathbb{R}^{{3^n}}$ is linearly independent

True or false: Every orthogonal set of vectors in $\mathbb{R}^{{3^n}}$ is linearly independent. I think that the answer is “false”, since an orthogonal set can contain the zero vector. But I'm not ...
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Orthogonal family of curves to the level sets of $f(x_1, \cdots, x_n) = \prod_{j=1}^{n} x_j$

I read this post and realized that if I can solve the related problem below that it might lead to an application. Their solution led to a hyperbola, so I suspect that my problem might lead to an $n$-...
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Orthogonal matrix with non-unit determinant

Can I have a matrix $Q$ which is orthogonal because each of the column vectors dot products with each other is 0? Or must only satisfy $QQ^T=I$. For example consider the following matrix $Q$: $$Q=\...
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Find all $f$ such that $\mathbb{E}[f(X)Y] = 0$

Let $X$ and $Y$ be random variables with finite mean and variance. We seek to find all measurable $f$ such that $\mathbb{E}[f(X) Y] = 0$, where $\mathbb{E}$ denotes the expectation with respect to $X$ ...
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Proving a General Finite Rotation in Index Notation is Orthogonal

We know that a general finite rotation by an angle $\phi$ about an axis with unit normal $n$ (such that $n_kn_k=1$) is given by the transformation $\bar{x_i}=A_{ij}x_j$ where the matrix $A$ is ...
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2 votes
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Orthogonal Matrix with respect to different inner products

From what I understand, an orthogonal matrix is one that satisfies $A\cdot A^t = I_n$. In such case, from what I saw online, $\forall x\in \mathbb{R}^n,\;\left\Vert Ax \right\Vert =\left\Vert x \right\...
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Using gram-schmidt to compute basis of column space.

Say you want to find a basis for the column space of a matrix. Can you apply Gram-Schmidt to the columns of A and take the output as your basis? What is the most efficient method in general for ...
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Is it possible to find the orthogonal complement of a single vector (something that is not a subspace)?

This is the problem that I have been asked: Find the orthogonal complement of the transpose of the vector = [3,4,1]. Also find the point on the plane 2x-3y+z=0 which is closest to (3,4,1). I know how ...
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$A=I_n+uu^t$ is symmetric and positive definite.

Let $A$ be a $n\times n$ matrix and let $u\neq 0 $ be a vector in $\mathbb{R}^n$ such that $$A=I_n+uu^t$$ ($t$ is the transpose). I was asked to show that $A$ is symmetric and positive definite. It ...
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