Questions tagged [inner-product]

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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 ...
S-F's user avatar
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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(...
Karim's user avatar
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2 answers
231 views

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 ...
Julian Newman's user avatar
3 votes
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101 views

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 ...
user3826143's user avatar
1 vote
1 answer
106 views

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 ...
GBA's user avatar
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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 ...
user3826143's user avatar
4 votes
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164 views

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 $\...
NIshant Rathee's user avatar
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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(...
mathoverflowUser's user avatar
0 votes
1 answer
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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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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^...
Javier Astorga's user avatar
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1 answer
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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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1 answer
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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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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 ...
John117's user avatar
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8 votes
1 answer
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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 ...
Markus's user avatar
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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 \...
Leo's user avatar
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1 answer
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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 $x \mapsto x+2\pi$). ...
nervxxx's user avatar
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2 votes
1 answer
131 views

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 ...
Chen's user avatar
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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. ...
Mira's user avatar
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0 answers
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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) ...
epsilon's user avatar
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2 answers
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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_\...
David Roberts's user avatar
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1 vote
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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 ...
Narnia's user avatar
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0 answers
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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>\...
Ben's user avatar
  • 1,010
6 votes
2 answers
293 views

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 ...
BinAcker's user avatar
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2 votes
0 answers
95 views

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,...
Timothy Chu's user avatar
0 votes
2 answers
1k views

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}} & &...
sleeve chen's user avatar
1 vote
1 answer
86 views

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 ...
Ali's user avatar
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2 votes
0 answers
270 views

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)\...
Ben's user avatar
  • 1,010
11 votes
1 answer
942 views

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, \...
Malkoun's user avatar
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113 votes
10 answers
73k views

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 (\...
4 votes
0 answers
128 views

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}\...
Alan Wiggins's user avatar
1 vote
0 answers
34 views

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 ...
DNQZ's user avatar
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3 votes
1 answer
868 views

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 ...
Mirco A. Mannucci's user avatar
2 votes
1 answer
327 views

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 \...
Prateek Sasan's user avatar
-1 votes
1 answer
279 views

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 (...
hookah's user avatar
  • 1,096
2 votes
1 answer
378 views

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 ...
Alec Edgington's user avatar
3 votes
1 answer
425 views

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 ...
user avatar
5 votes
2 answers
618 views

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 \...
Pierre Dubois's user avatar
4 votes
1 answer
829 views

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 ...
B Merlot's user avatar
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1 vote
1 answer
2k views

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 ...
Cogicero's user avatar
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17 votes
2 answers
504 views

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 ...
user521337's user avatar
  • 1,189
2 votes
1 answer
173 views

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 $$ \...
Ju Sun's user avatar
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2 votes
2 answers
244 views

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 $\mathbb{R}$. ...
user173856's user avatar
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1 vote
1 answer
268 views

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 ...
Saj_Eda's user avatar
  • 395
6 votes
1 answer
271 views

Which metrics on exterior power are induced from metrics on the base?

$\newcommand{\id}{\text{id}}$ $\newcommand{\Hom}{\text{Hom}}$ This is a cross-post. Let $V$ be a $d$-dimensional real vector space, and let $2 \le k \le d-1$. Every inner product on $V$ induces an ...
Asaf Shachar's user avatar
  • 6,591
4 votes
1 answer
401 views

On a vanishing integral inner product

Let $G(z)$ be an $n\times m$ rational matrix-valued function of full column rank on the unit circle. Further, let $P(z)$ be an $m\times m$ rational matrix-valued function positive definite on the unit ...
Ludwig's user avatar
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2 votes
0 answers
88 views

System of bilinear equations [closed]

Given constants $n$ and $\ell$, where $\ell < n$, suppose we have the following set of equations in column vectors $a_1, a_2, \dots, a_n \in {\mathbb{C}^{\ell}}$ $$\begin{array}{rl} \langle ...
Tony Yan's user avatar
3 votes
0 answers
1k views

Inner Product on tensor product of Hilbert spaces is unique?

Given two Hilbert Spaces $H$ and $K$, a natural inner product on $H\otimes K$(= vector space tensor product of $H$ and $K$) is given by $\hspace{.5in}\langle h_1\otimes k_1,h_2\otimes k_2\rangle=\...
Manish Kumar's user avatar
2 votes
0 answers
128 views

Do involutions always stabilize some transverse lagrangians?

Let $V$ be a vector space of dimension $2n\geq 4$ over a field $F$ of characteristic distinct from $2$. Assume that $V$ is equipped with a nondegenerate alternating form $b$. Let $Sp(V)$ denote the ...
Oliver's user avatar
  • 367
13 votes
1 answer
675 views

Vectors that are almost orthogonal on average: lower bounds on dimension?

Let $v_1,\dotsc,v_k \in \mathbb{R}^d$ be unit-length vectors such that $$\sum_{1\leq i,j\leq k} |\langle v_i,v_j\rangle|^2 \leq \epsilon k^2.$$ What sort of lower bound can we give on $d$ in terms of $...
H A Helfgott's user avatar
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3 votes
0 answers
190 views

Probability of orthogonal vectors?

Denote $\mathcal I_m=\{-m,-m+1,\dots,0,\dots,m-1,m\}$ where $m\geq0$ is an integer. Pick a uniformly random vectors $$\hat a=(a_n,a_{n-1},\dots,a_1)\in(\mathcal I_{m_1}+\sqrt{-1}\cdot\mathcal I_{m_2}...
Turbo's user avatar
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