Questions tagged [orthogonality]

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

Filter by
Sorted by
Tagged with
0 votes
0 answers
22 views

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 ...
Martin's user avatar
  • 1
0 votes
0 answers
25 views

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 ...
TobiR's user avatar
  • 478
0 votes
0 answers
39 views

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) \...
jason 1's user avatar
  • 513
1 vote
0 answers
44 views

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:/...
Ans shakeel's user avatar
5 votes
3 answers
71 views

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 ...
Shuri2060's user avatar
  • 4,313
0 votes
0 answers
41 views

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. ...
whisdom's user avatar
0 votes
1 answer
49 views

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, ...
Important_man74's user avatar
1 vote
1 answer
26 views

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 $...
clay's user avatar
  • 2,609
0 votes
0 answers
22 views

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 ...
Shivam Vishwekar's user avatar
0 votes
1 answer
63 views

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), ...
Jordan's user avatar
  • 123
3 votes
1 answer
55 views

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. ...
Phoenix's user avatar
  • 175
5 votes
1 answer
59 views

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 ...
Ofek Tevet's user avatar
0 votes
0 answers
24 views

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_{...
TheDude's user avatar
  • 151
1 vote
1 answer
47 views

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{...
MackeyTopology's user avatar
1 vote
1 answer
37 views

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 ...
tisPrimeTime's user avatar
1 vote
1 answer
59 views

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 ...
Mud's user avatar
  • 61
1 vote
0 answers
32 views

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 ...
skm's user avatar
  • 113
1 vote
1 answer
27 views

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 ...
MegaFish TV's user avatar
2 votes
1 answer
71 views

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 ...
cdahms's user avatar
  • 123
2 votes
0 answers
40 views

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 ...
babu's user avatar
  • 85
3 votes
1 answer
32 views

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 = \...
matdlara's user avatar
  • 321
1 vote
1 answer
39 views

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 ...
babu's user avatar
  • 85
1 vote
0 answers
69 views

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 ...
babu's user avatar
  • 85
0 votes
1 answer
39 views

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 $v \perp w \implies Av \perp Aw$. ...
tudale's user avatar
  • 3
0 votes
1 answer
40 views

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(\...
AnonA's user avatar
  • 43
0 votes
0 answers
15 views

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 ...
JC Olivier's user avatar
1 vote
2 answers
130 views

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)...
Andromeda's user avatar
  • 356
2 votes
0 answers
42 views

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 ...
Marius Lutter's user avatar
1 vote
1 answer
78 views

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 ...
SRobertJames's user avatar
  • 2,722
3 votes
1 answer
62 views

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 ...
yeseong Bae's user avatar
0 votes
1 answer
29 views

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 ...
lll's user avatar
  • 1
0 votes
1 answer
44 views

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: ...
Jason Curran's user avatar
1 vote
0 answers
47 views

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. ...
user129393192's user avatar
2 votes
1 answer
60 views

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 ...
user129393192's user avatar
0 votes
1 answer
73 views

$SAS^{-1} = B$, $S,A \in \operatorname{Mat}_{3,3}(\mathbb{R})$ 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} \\-...
Marius Lutter's user avatar
0 votes
0 answers
29 views

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 ...
matdlara's user avatar
  • 321
2 votes
0 answers
44 views

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 $\...
YSA's user avatar
  • 113
1 vote
0 answers
24 views

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 ...
SebastianP's user avatar
2 votes
1 answer
39 views

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 $...
zxcv's user avatar
  • 1,363
0 votes
0 answers
37 views

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 ...
Viuツ's user avatar
  • 43
0 votes
0 answers
15 views

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 ...
wanymose's user avatar
  • 552
1 vote
0 answers
31 views

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 ...
WomBud's user avatar
  • 13
0 votes
0 answers
21 views

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 ...
1g2b3c4d's user avatar
3 votes
1 answer
63 views

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 ...
MathStudent101's user avatar
2 votes
2 answers
244 views

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 \...
anyon's user avatar
  • 329
0 votes
4 answers
110 views

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 ...
nonaaas2's user avatar
0 votes
0 answers
36 views

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 ...
wanymose's user avatar
  • 552
0 votes
0 answers
65 views

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 ...
gm01's user avatar
  • 69
0 votes
1 answer
45 views

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)...
Lose' CKi's user avatar
0 votes
1 answer
61 views

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 $...
Quantumwhisp's user avatar

1
2 3 4 5
53