Questions tagged [orthogonality]
This tag can be used to refer to the orthogonality of a set of vectors, a matrix or a linear transformation.
2,602
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How to make the gradient for a negative in triplet loss move ortogonal to the anchor and positive?
So i've implemented a convolutional neural network in C++ and I'm playing around with loss functions and gradient calculations. The network outputs the f(a), f(p) and f(n) of the last layer, which is ...
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How to choose a vector which is linearly independent from a set of orthogonal vectors?
I have a non-complete set of orthogonal vectors $V=[\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_n]$ with $m > n$ entries.
I would like to choose another vector $\mathbf{w}$ which is linearly ...
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For orthogonal matrix $P = [P_1 \vert P_2]$, show that $col(P_1)^{\bot} = col(P_2)$
I'm currently struggling to solve this question.
The first one $col(P_2) \subset col(P_1)^{\bot}$ is quite straightforward
($P_1$ is ($n \times r$) and $P_2$ is ($n \times (n-r)$))
$x \in col(P_2) \...
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Calculate Orthonormal Basis by Using only one vector (direction vector)
Goal: Drawing a Set of circular point around a center point in 3D space facing towards the direction vector. End Goal is to draw a Cylinder and Bend.
Followed this approach to draw a circle 3D: https:/...
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Generalized notion of perpendicularity, (not orthogonal)
In 3 dimensions, we might call 2 planes perpendicular iff their normals are orthogonal.
But this does not coincide with the definition of orthogonal subspaces - the dot product of any pair of vectors ...
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Computing conditional expectation without densities
I am currently looking into conditional expectations (CE) and how to compute them. I know that in the typical finite-dimensional, continuous settings, one would do the computations via densities. ...
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Find a set of orthogonal vectors which are still orthogonal under an operator.
Let $V$ be an $n$-dimensional real inner-product space and let $T$ be a linear operator on $V$. Prove that there exists a nonzero orthogonal list $(v_1,v_2,\dots, v_n)$ such that $(Tv_1, Tv_2, \dots, ...
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Show $A \subset B \Leftrightarrow B^{\perp} \subset A^{\perp}$ with $A,B$ subsets of a Hilbert space.
Show $A \subset B \Leftrightarrow B^{\perp} \subset A^{\perp}$ with $A,B$ subsets of a Hilbert space.
The forward direction is easy: Assume $A \subset B$. For any $a \in A$ and $x \in B^\perp$ since $...
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A Problem on Family of circles and family of corresponding orthogonal circles
Problem Statement:
Let $P(x_1,y_1)$ and $Q(x_2,y_2)$ be two fixed points on xy plane and $R(\alpha , \beta)$ is a point such that $PR:QR=k , \ (k≠1)$ and locus of R for different values of k be ...
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If a set of vectors are all orthogonal to each other, would shortened versions of those vectors also be orthogonal? [closed]
Say I have a set of vectors, $v_{0\ldots n}$, all of which are orthogonal to each other and all of which are of length $m$.
If I took a portion of each vector (same start and end indices for each), ...
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Angle between column vectors after centring a semi-orthogonal matrix
Given a set of $n$, $m$ dimensional orthonormal vectors $\mathbf x_1, ..., \mathbf x_n$, where $m \geq n$. Let $\boldsymbol\mu = (\mu_1, ..., \mu_m)$ be the means of the vectors along each dimension. ...
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Spectral theorem for diagonal matrix in different inner product spaces
I learned a special case of the spectral theorem for finite dimensional inner product space. As I understand it states that a real matrix is orthogonally diagonalizable with real eigenvalues iff it ...
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The Inner Product of a Hadamard Product
So let's say I have the inner product:
$$\vec{y}_1^H \vec{y}_2 = (\vec{x}\circ\vec{h}_1)^{H} (\vec{x}\circ\vec{h}_2) = \sum_{i} (x_i^{\ast} h_{1,i}^*) (x_i h_{2,i}) = \sum_{i} |x_i|^2 h_{1,i}^* h_{...
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Is a orthogonal projection in a Hilbert space automatically selfadjoint?
Let $P$ be a self adjoint projector on a Hilbert space $H$ i.e. $P: H \rightarrow H$ is linear and continuous, $P^*=P$ and $P^2=P$
Then $P$ is also an orthogonal projection i.e. $\mathrm{ran}P=\mathrm{...
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Dimensionality not matching for differential of matrix with orthogonality constraints
I was reading through the following answer out of curiosity about calculating the differential of a matrix with orthogonality constraints. Briefly the mathematics works out as follows:
$
\text{Let } X ...
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Property of Orthogonal Projections
Let $H$ be a real Hilbert space with inner product $(\cdot, \cdot)$. Assume that there is a subset $C \subset H$ that satisfies
$$C=\{x \in H : (x,y) \geq 0\} \forall y \in C$$
$C$ can be shown to be ...
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How come Eccentricity and Polar angle are orthogonal dimensions?
I am trying to understand the process of creating Visual Field Maps (VFM, also knows and Retinotopy) using fMRI with respect to the two orthogonal dimensions in visual space: eccentricity and polar ...
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Orthogonal functions and Riemann sums
Define inner product as $\langle f, g \rangle_{[a,b]} := \int_a^b f(x)g(x) \ dx$.
Say $f,g$ are orthogonal: $\langle f, g \rangle_{[a,b]} = 0 \Leftrightarrow \int_a^b f(x)g(x) \ dx = 0 \Leftrightarrow ...
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orthogonal vs orthonormal matrices - what are simplest possible definitions and examples of each ??
I'm trying to understand orthogonal and orthonormal matrices and I'm very confused. Unfortunately most sources I've found have unclear definitions, and many have conflicting definitions!
Some sites ...
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Kernel Polynomials and Extremality
I am trying to prove the following theorem:
Theorem Let $x_0 \in \mathbb{R}$ and $q_n(x)$ a polynomial of degree at most n, normalized by the following condition:
$\int_a^b (q_n(x))^2 w (x)dx$=1.
The ...
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normal operator where the sum of eigenspaces is not equal to the entire Hilbert space.
Hi I'm reading about Hilbert spaces and normal operators. In this i found the following result:
Let $H$ a Hilbert space of finite dimension and $T\in\mathcal{L}(H)$ a normal operator, then
$$
H = \...
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Proving the reproducing property of kernel polynomials
I need to prove the following property related to kernel polynomials:
$\int_a^b K_n(t,x)q_n(x)w(x)dx=q_n(t)$,
where $q_n(x)$ is a polynomial of degree less or equal to $n$, $w(x)$ is a weight function ...
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Finding the minimum value of an integral using least squares-mean function approximation
I am starting to study some concepts related to orthogonal polynomials and my teacher told me to prove the following theorem,
Theorem
The integral $\int_{a}^{b} Q_n^2(x)w(x) dx$ where $Q_n(x)$ is any ...
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There is a vector whose orthogonal complement is preserved under a linear map.
The problem:
Prove that for any non-zero $n$-dimensional matrix $A$ there is a non-zero vector $v \in \mathbb R^n$ such that for every $w \in \mathbb R^n$ we have $v \perp w \implies Av \perp Aw$.
...
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Orthogonal vector to ellipsoid surface is... $\vec{0}$?
I was looking at this ellipsoid:
$$
\frac{x^2}{25}+\frac{y^2}{25}+\frac{z^2}{9}=1
$$
I tried parametrizing it as such:
$$
\gamma\left(\theta, \varphi\right)=\left(5\cos\left(\theta\right)\sin\left(\...
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Given a binary random matrix A, M rows & N columns, how many non-zero binary column vectors b (Z 1's, Z <<N) exist so mod(A*b,2) = all zero vector?
The matrix $\mathbf A$ is a binary random matrix, M by N. One's and zeros equally likely. $\mathbf A$ is a full rank matrix. I form the GF2 product mod($\mathbf A \mathbf b,2$) with a binary column ...
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Element of minimal distance to a convex, closed cone: orthogonality?
I'm trying to prove the following theorem:
Let $H$ be a Hilbert space and $C\subseteq H$ a convex, norm-closed cone. Let $\xi \in H$. There is a unique $\eta \in C$ such that $\|\xi-\eta\| = d(\xi, C)...
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A basis of a diagonalized symmetric bilinear form
The symmetric bilinear form $b$ on $\mathbb{Q}^{3}$ is represented by the matrix with respect to the standard basis
$$A=\left(\begin{array}{ccc} 3 & -2 & 0 \\ -2 & 2 & -2 \\ 0 & -2 ...
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What is the locus of points in the plane $\{v : v \cdot (v-a) = 0\}$ for fixed $a$?
Fix $a$ in $\mathbb R^2$. What is the locus of points $\{v : v \in \mathbb R^2$ and $v \cdot (v-a) = 0\}$?
Clearly, $0$ and $a$ are in this set, and no other multiple of $a$ is in the set. Beyond ...
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Matrix representation of adjoint operators
I am studying linear algebra.
In the textbook I use, the book says that(Friedberg, 5th ed, p.356 theorem 10)
If $V$ is an f.d.i.p.s and $B$ is an O.N.B for $V$, $[T^{*}]_{B}=[T]_{B}^{*}$ holds for ...
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Orthogonal projections and Orthogonal Complements
I'm reading on orthogonal projections from a course's notes and it says the following:
For each x ∈ $R_n$ and each linear subspace U, $\pi_U$(x) exists and is unique. Moreover, $\pi_U$(x) is the only ...
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Clarification of terms with reference to weight function of Orthogonal Polynomials.
I am looking at "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables" by Abramowitz and Stegen. In particular I am looking at the beginning of Chapter 22: ...
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The concept of a linear map being self-adjoint
In class, we were told:
let V be a finite dimensional inner product space. A linear map $T: V \to V$ is self if $T = T^*$
Later, we were told:
let V be a finite dimensional real inner prod. space. ...
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A linear map with an orthonormal eigenbasis
Here is a proposition my professor stated in class:
Let V be a finite dimensional inner product space, with $ T: V \to V$ a linear map. Let $\gamma$ be an orthonormal basis of eigenvectors of V.
The ...
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$SAS^{-1} = B$, $S,A \in \operatorname{Mat}_{3,3}(\mathbb{R})$ but $A$ has complex eigenvalues
Consider the following matrix $A \in \operatorname{Mat}_{3,3}(\mathbb{R})$:
$$A=\frac{1}{90}\left(\begin{array}{ccc} 66 & -18 \sqrt{6} & 30 \sqrt{2} \\6 \sqrt{6} & 72 & 30 \sqrt{3} \\-...
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Relations of injective and surjective in adjoint opeerators.
I'm trying to solve the converse of following excercise
where E and F are Banach Space, $E^*$ and $F^*$ are the dual space, $S(E,F)$ is the set of surjective linear and bounded maps and $I(E,F)$ is ...
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Inner product with two different bases
Let $V$ be a finite-dimensional inner product space. If $B= \{ b_{1}, b_{2},\cdots, b_{n}\}$ is a basis for $V$, show that $B'=\{f_{1},f_{2},\cdots,f_{n}\}$ is also a basis for $V$ with property $\...
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Orthonormal functions for squared Mathieu functions
I am working with Laplace-Eigenvalues of ellipses and in this context I started working with Mathieu functions. Now I have reached a point where I can no longer go any further.
I am currently looking ...
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How to extend an orthogonal linearly independent set to an orthogonal basis, with respect to a symmetric bilinear form?
Let $F$ be a field whose characteristic is not $2$. Let $X$ be an $n(<\infty)$-dimensional vector space over $F$, equipped with a symmetric bilinear form $\langle\cdot,\cdot\rangle$ on $X$. Let $...
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Calculating the orthogonal projection for polynomial functions
The task is:
Let ${E := {ax^3+bx^2+cx+d : a, b, c, d ∈ R}}$ be the vector space of
all real polynomials of degree at most 3 and let ${F}$ be the
subvector space of all real polynomials of degree at ...
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Question about prove of characterization of projection [duplicate]
I am trying to prove that:
Let $K$ be a closed konvex subset of a Hilbert space $H$. Let $x_o \in H$, then for all $x \in K$ the following are equivalent:
$\lVert x_o -x \rVert = inf_{y \in K} \lVert ...
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self-adjoint projection
for the following exercise i have some questions, i appreciate some help or hints for this exercise.
Let be V an euclidean or unitary vectorspace and $ p:V \to V$ a self-adjoint Projection.
$(a)$ Show ...
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Calculate Orthogonality of 2 Identical Curves
I have the following question
I have two identical functions:
f(x)= -cosh(x)
g(x)= f(x)
How much do I have to move g(x) along the x-axis, so that f(x) and g(x) create a 90° angle between the two ...
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Let $T\colon V\to V$ over the IPS $V$ and $B$ be an orthogonal basis for $V$. Find the simplest connexion between $[T^*]_B$ and $([T]_B)^*$
Let $T\colon V\to V$ over an Inner Product Space $V$.
Let $B$ be an orthogonal basis for $V$.
Find the simplest connexion between
$[T^*]_B$ and $([T]_B)^*$.
So I know that if $B$ was an orthonormal ...
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Equivalent of sequence convergence in Hilbert space
$H$ is a Hilbert space, $\{x_{n}\}$ is a orthogonal family in $H$(not need to be Orthonormal), prove that the following conditions are equivalent:
$\sum_{n=1}^{\infty} x_{n}$ convergence
$\forall y \...
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Find values of $k$ for which the lines $3x - ky = 5$ and $(k^2 - 2)x + 3y = 4$ are perpendicular
Problem:
I'd like to find the values of $k$ for which the two lines
$$3x - ky = 5\quad\text{and}\quad (k^2 - 2)x + 3y = 4$$
are perpendicular.
I have been trying to solve this problem for hours, and I ...
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Using Hilbert projection theorem to prove properties of projection map
First consider the following theorem:
Let $H$ be a hilbert space and $S$ be a closed subspace of $H$.
$H=S \oplus S^{\perp}$, i.e. for $x \in H$ there exists (only one) representation of $x$ as $x=s+y ...
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Hausdorff Distance Between Orthogonal Complements
Let $H$ be a finite-dimensional complex Hilbert space and denote by $d_{\textrm{Haus}}$ the Hausdorff distance between linear subspaces of $H$ i.e., $d_{\textrm{Haus}}(V,W)$ is the usual Hausdorff ...
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Gaussian random matrix rotation invariance
Checking exercise 3.3.3 from R. Vershynin
Let G be an m × n Gaussian random matrix, i.e. the entries of G are
independent N(0, 1) random variables. Let u ∈ Rn be a fixed unit vector.
Then Gu ∼ N(0, Im)...
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If I have complete set of vectors that are orthogonal with respect to a matrix, does that mean they are eigenvectors of that matrix?
I have a symmetric full rank matrix M, and a set of vectors $v_i$ that span the whole space. I assume those vectors to be orthogonal (they form an orthogonal set).
If two of those vectors $v_i$ and $...
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