Questions tagged [inner-product]
The inner-product tag has no usage guidance.
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abc-conjecture and positive definite kernels, again?
One formulation of the abc-conjecture is:
$$\forall a,b \in \mathbb{N}: \frac{a+b}{\gcd(a,b)}< \operatorname{rad}\left ( \frac{ab(a+b)}{\gcd(a,b)^3}\right )^2 $$
Let us define:
$$K(a,b) := \frac{2(...
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What are alternative or equivalent definitions of a positive-definite function on a group?
The standard definition of a positive-definite function on a group goes as follows:
Let $\varphi : G \rightarrow L(H)$, where $G$ is a group (with an involution) and $H$ a Hilbert space. $L(H)$ is the ...
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The discrete orthogonal polynomials
I want a document or something that explains the following proposition:
The discrete orthogonal polynomials are the polynomial solutions of the given diference equation:
$$
\sigma(x)\Delta\nabla P_n(...
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Does "perpendicular phase incoherence" satisfy the triangle inequality?
I asked this question at https://math.stackexchange.com/q/4783968/222867, but even after a 200-point bounty, no solution was provided, only some thoughts regarding possible directions. So I'm now ...
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Definition of inner product for vector spaces over arbitrary fields
Is there a canonical definition of the concept of inner products for vector spaces over arbitrary fields, i.e. other fields than $\mathbb R$ or $\mathbb C$?
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Unitary operators with the same inner product as vectors
Suppose we have a set of real unit vectors $v_1,\ldots,v_m \in \mathbb{R}^n$. We can always find a set of unitary operators $U_1,\ldots,U_m$ acting on $\mathbb{C}^N$ (for $N$ that is possibly much ...
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An inner product and projection property in RKHS
I'm reading an article regarding the condition for the existence of a solution to a constrained Pick interpolation problem, and the author wrote the following equality - for background, we have the ...
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Uniform distribution on pairs of unitary matrices
This question has two parts.
In Part 1, I would like to know if the following distribution on pairs of $d$-dimensional unitary matrices has popped up in the literature:
"Uniform distribution on ...
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What is the exponential map from the Lie algebra $\mathfrak{sl}(2,\mathbb{C})\ltimes_\textrm{ad}\mathfrak{sl}(2,\mathbb{C})$ to its Lie group?
$\DeclareMathOperator\ad{ad}\DeclareMathOperator\Ad{Ad}\DeclareMathOperator\Exp{Exp}\DeclareMathOperator\SL{SL}\DeclareMathOperator\sl{\mathfrak{sl}}$Let $G:=\SL(2, C) \ltimes_{\Ad} \SL(2,C)$, where $\...
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Distribution of inner product of random unit vectors
Suppose I have two $2\times2$ Haar random unitary matrices $u_1$ and $u_2$, then I can define a diagonal matrix $$\begin{pmatrix}(u_1\cdot u_2)_{11}&0\\ 0&(u_1\cdot X\cdot u_2)_{11}\end{...
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Maximum number of vectors with bounds on inner products (follow up question)
This is a follow-up question from my previous question.
Suppose there are (2n+1) vectors $\{m_1,m_2,...,m_n\}$, $\{p_1,p_2,...,p_n\}$ and $p^*$ in $R^{k+1}$. $m_i$ are weakly positive vectors. $p_i$ ...
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Maximum number of vectors with bounds on inner products
Suppose there are 2n vectors $\{m_1,m_2,...,m_n\}$ and $\{\mu_1,\mu_2,...,\mu_n\}$. All vectors are in k-dimensional Euclid space $R^k$. The vectors satisfy:
$$m_i \mu_j \leq 0, \quad \forall i\neq j $...
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What fields can be used for an inner product space?
The title is the question: What fields can be used for an inner product space?
This question has been discussed in Math Stack Exchange with no definitive resolution. A similar question appeared here, ...
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Inner product over finite field
sorry for informals but is my first post.
In Coding theory (exactly in Coding theory a first course - San ling and Chaoping) found this definition:
$\langle , \rangle :\mathbb{F}_q^n\times\mathbb{F}_q^...
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Invariance signature in infinite dimension
Let $V$ be an infinite dimensional vector space and suppose we have a smooth family $\{g_t\}_{t\ge 0}$ of symmetric bi-linear forms such that:
$g_0$ is positive-definite
$g_t$ is non-degenerate for ...
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Inner products on super vector spaces
Let $V=V^0\oplus V^1$ be a super vector space (https://en.wikipedia.org/wiki/Super_vector_space)
Is there a special definition of an inner product on $V$ other than just an inner product on the ...
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Parallelogram law for vectors of equal length
Does the parallelogram law for vectors of equal length imply the full parallelogram law? That is,
if for all norm one vectors $x$ and $y$ in a Banach space $X$ it holds that $\lVert x-y\rVert^2+\lVert ...
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Unimodular eigenvalue of a H-self-adjoint matrix (indefinite innerproduct)
Let $A,H \in \mathbb{C}^{n \times n}$ be such that $H$ is Hermitian and invertible and $A = H^{-1} A^* H$. In this case, $A$ is said to be $H$-self-adjoint. This is due to the fact that if $\langle \...
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Orthogonal functions on circle with constraints
I have a curious question I stumbled upon that may be interesting to some.
Consider real-valued continuous functions on the circle $f_1(x),f_2(x),f_3(x)$ (so they are periodic in $x \mapsto x+2\pi$).
...
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Largest number of two series vectors with negative dot product
Assume $\{x_{i}\}_{i=1}^{m}$, $\{w_{i}\}_{i=1}^{m}$ are two sets of vectors in $\mathbb{R}^{n}$. And we have that $ x_{i}\cdot w_{j} < 0$ for $i \neq j$ and $x_{i}\cdot w_{i} > 0$ for all $i$. I ...
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The inner product of a Clifford Algebra
Any Clifford algebra $\operatorname{Cl}(k, p)$ carries an induced inner product, which is the "trace" on its 0-blade: $\langle AB\rangle_0$ for given elements $A, B$ of the algebra.
This inner ...
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Question about finite dimensional representations of a semi-simple Lie group
I have posted a question in MSE
https://math.stackexchange.com/questions/4468138/question-about-finite-dimensional-representations-of-a-semi-simple-lie-group but didn't receive any comment or answer.
...
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What is lost after RKHS embedding of the L1 space?
We know that every distribution or $L^1$ function $f$ over space $\mathcal{X}$ (e.g., $R^d$) can be embedded to an RKHS $\mathcal{H}$ with a $1$-bounded kernel $\mathcal{K}$ (e.g., the RBF kernel) ...
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What is Pressley and Segal's "basic inner product" for compact simple Lie algebras of types B and C?
In Pressley and Segal's book Loop Groups, they define a "basic inner product" $\langle-,-\rangle$ on a simple Lie algebra to be (minus) the Killing form scaled so that $\langle h_\alpha,h_\...
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The existence of a variety generated in low degree that is epsilon-close to a fixed variety
Let $\mathbb{C}^n$ be the $n$-dimensional complex vector space endowed with the standard Hermitian inner product, let $X \subseteq \mathbb{C}^n$ be an algebraic set that forms a cone, and let $1>\...
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General bivariate functions that satisfy Cauchy-Schwarz
Have there been any study of general bivariate functions $f:X \times X \to \mathbb{R}$ that satisfy $f(x,y)^2 \leq f(x,x)f(y,y)$. This comes up as a function I'm working with satisfies the asymmetric ...
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Any inequalities / estimates for a lower bound of the $L^2$ inner product of a quantity and its derivative?
For some numerical analysis of a fluid, I am wondering if there is any inequality that provides a lower bound (in the $L^2$ norm), for the $L^2$ inner product of a quantity with its derivative. In ...
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Kernels with finite dimensional feature spaces
Suppose $x,y \in \mathbb{R}^n$ for some given fixed n.
Consider a kernel $K(x,y) = f(\langle x, y \rangle)$, I'd like to know which functions $f$ admit a finite dimensional feature map. In other words,...
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Matrix norm minimization and matrix inner product
One of the famous problem in SDP is the matrix norm minimization (see S. Boyd, Convex Optimization, p. 170).
Consider:
\begin{equation}\label{eq:Lasse}
\begin{aligned}
&\min_{\mathbf{x}}
& &...
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A linear algebra question for semi-Euclidean norm
Let us consider Minkowski inner product on $\mathbb R^{1+n}$, defined by
$$ \langle v,w \rangle = -v_0w_0+\sum_{j=1}^n v_j w_j\quad \,\forall\, v,w \in \mathbb R^{1+n}.$$
We say that a vector $v$ is ...
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Why do inner products require conjugation?
For Hermitian matrices and operators, the most "natural" inner product is $f^H \cdot g$ or $\int f^* g\; dx$. A similar situation holds interpreting Fourier transforms as the inner product of ...
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What are the benefits of writing vector inner products as $\langle u, v\rangle$ as opposed to $u^T v$?
In a lot of computational math, operations research, such as algorithm design for optimization problems and the like, authors like to use $$\langle \cdot, \cdot \rangle$$ as opposed to $$(\cdot)^T (\...
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Is there a name for this "inner product" on projective space?
$\newcommand{\proj}{\mathbb{P}}\newcommand{\complex}{\mathbb{C}}\newcommand{\ip}[2]{\langle #1 , #2\rangle}\newcommand{\abs}[1]{\lvert #1 \rvert}$There is a natural bijection $\phi: \proj(\complex^n)\...
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Higher order generalization of Cauchy-Schwarz?
Is there a generalization of the Cauchy-Schwarz inequality along the following lines? Let $V$ be an inner product space (for simplicity of notation, let us work over the real numbers). Let $v_1, \...
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Independent inner functions on the unit disk
This problem cropped up in a paper that I am writing and I have thought about it for too long to no avail: let $\mathbb{D}$ be the open unit disk in the complex plane and suppose $\varphi:\mathbb{D}\...
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Can an orderless set of inner product between N vectors determine unique structure of the vectors?
Suppose we have n vectors {a1,a2,a3,...,an} such that the sum of them is zero vector
a1+a2+a3+...+an=0
Now, we compute the inner product of each two vectors of them, i.e. we compute the Gramian matrix ...
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Bound for matrix inner product based on singular values
Regarding the matrix inner product based on singular values, Lewis (1995) "The convex analysis of unitarily invariant matrix functions" states the result by von Neumann that $\langle X,Y \rangle \leq \...
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Expressing the sum of two squared inner products more compactly: is it possible to lift the dimension? [closed]
Let $v_1,v_2\in\mathbb{R}^d$ be two fixed vectors, and $\langle \cdot,\cdot\rangle_{\mathbb{R}^d}$ be the usual Euclidean inner product in $\mathbb{R}^d$.
My question is as follows. Is there an (...
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Mean squared absolute value of inner product of unit vectors
Given a nonempty finite subset $S$ of the unit sphere of $d$-dimensional complex Hilbert space, let $\lambda(S) = \frac{1}{\lvert S \rvert^2} \sum_{x,y \in S} \lvert \langle x,y \rangle \rvert^2$ be ...
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Linearity of the inner product using the parallelogram law
A norm on a vector space comes from an inner product if and only if it satisfies the parallelogram law. Given such a norm, one can reconstruct the inner product via the formula:
$2\langle u,v\rangle ...
CommunityBot
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Neat/Approximate formula for maximum number of "almost orthogonal" vectors in a complex vector space?
In a $d$ dimensional vector space defined over $\mathbb{C}$, how do I calculate the largest number $N(\epsilon, d)$ of vectors $\{V_i\}$ which satisfies the following properties. Here $\epsilon$ is ...
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Non-standard tensor products of inner product spaces
For two inner product spaces $(\mathcal{V}, (\cdot,\cdot)_V)$ and $(\mathcal{W}, (\cdot,\cdot)_W)$, we can put an inner product on their tensor product in the obvious way:
$$
(1) ~~~~ \langle v \...
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Which inner products preserve positive correlation?
Suppose we have a symmetric PD or PSD matrix M which induces an inner product $\langle \cdot, \cdot \rangle_M$. If we have that $\langle x, y \rangle > 0$ for two unit vectors $x$, $y$, are there ...
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On a special type of normed linear spaces
Let $(V,\|.\|)$ be a normed linear space such that for every group $(G,*)$, every function $f:G \to V$ satisfying
$$
\|f(x*y)\|\ge \|f(x)+f(y)\|,\qquad\forall x,y\in G,\tag{Z}
$$ is a group ...
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Minimum off-diagonal elements of a matrix with fixed eigenvalues
I am an engineer working in radar research. I came accross a problem on which I cannot seem to find literature. I can ask it in two different ways. Perhaps depending on the reader, the alternative ...
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Controlling angles between vectors using sum of subvector angles?
This is a technical question coming out of my research.
Let $\angle(\cdot, \cdot)$ be the angle ($\in [0, \pi]$) between vectors. Consider two vectors $u, v$ in $\mathbb R^3$. Is it true that
$$
\...
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Hash functions and inner product
As part of a research project on derandomization of linear threshold functions I am working on, I am trying to understand the following problem:
Is there a small (polynomial rather than exponential)...
CommunityBot
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Almost orthogonal vectors in subsets of euclidean space
Given the vector space $\mathbb{R}^n$, endowed with the standard inner (dot) product $\langle\cdot,\cdot\rangle:\mathbb{R}^n\times \mathbb{R}^n\to\mathbb{R}$, the problem of almost-orthogonal sets ...
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Adjoint of an operator-valued linear operator
I have come across a linear bounded operator $B:K\to \mathcal{L}(U,Z)$ where $K$, $U$, and $Z$ are separable Hilbert spaces. I need a reference (any source) to find out about:
The adjoint of such an ...
CommunityBot
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An inner product on the vector space $\mathbb{R}[x_1,\cdots,x_n]_m$
For any given integers $m,n\geq1$, let $\mathbb{R}[x_1,\cdots,x_n]_m$ be the vector space of homogeneous polynomials of degree $m$ in $x_1,\cdots,x_n$ over the field of real numbers $\mathbb{R}$. ...
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