Questions tagged [inner-product]

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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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0 votes
0 answers
42 views

Concentration inequality for inner product of two Lipschitz functions

I was reading chapter 5 of the book HDP(Roman Vershynin). There I find theorem 5.1.4 extremely fascinating. I am curious to discover does this theorem hold to the inner product of two Lipschitz ...
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8 votes
1 answer
391 views

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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2 votes
0 answers
40 views

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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1 vote
1 answer
156 views

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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2 votes
1 answer
93 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 ...
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2 votes
0 answers
73 views

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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1 vote
0 answers
54 views

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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5 votes
2 answers
284 views

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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1 vote
0 answers
43 views

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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3 votes
0 answers
121 views

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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5 votes
2 answers
184 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 ...
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2 votes
0 answers
69 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,...
0 votes
2 answers
611 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}} & &...
1 vote
1 answer
81 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 ...
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2 votes
0 answers
186 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)\...
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11 votes
1 answer
860 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, \...
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106 votes
10 answers
65k 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
107 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}\...
1 vote
0 answers
33 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 ...
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3 votes
1 answer
460 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 ...
2 votes
1 answer
212 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 \...
-1 votes
1 answer
142 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 (...
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2 votes
1 answer
257 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 ...
2 votes
1 answer
327 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 ...
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5 votes
2 answers
580 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 \...
4 votes
1 answer
573 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 ...
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1 vote
1 answer
1k 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 ...
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17 votes
2 answers
475 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 ...
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2 votes
1 answer
164 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 $$ \...
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2 votes
2 answers
218 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 $\mathbb{R}$. ...
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1 vote
1 answer
265 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 ...
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6 votes
1 answer
235 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 ...
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4 votes
1 answer
372 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 ...
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2 votes
0 answers
85 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 ...
3 votes
0 answers
984 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=\...
2 votes
0 answers
125 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 ...
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13 votes
1 answer
632 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 $...
3 votes
0 answers
162 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}...
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0 votes
0 answers
256 views

Inner products on abelian groups and general modules

Where can I find a discussion about inner products on something more general than vector spaces, and to which extent should one attempt such a generalization? My particular interest is in abelian ...
4 votes
0 answers
105 views

Triple Petersson Inner Products With Theta Functions

Our current work requires us to bound triple products of the form $$\langle \Im(*)^{\frac{3}{2}}|\theta^3|^2,\mu_j \rangle,$$ where $\langle \cdot, \cdot \rangle$ is the Petersson inner product, ...
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1 vote
0 answers
34 views

Minimum element estimation of A*A^T [closed]

I want to estimate the minimum element of A*A^T using only the information of A. For example, maximum element can be estimated using the norm values of all rows without calculating all elements of ...
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9 votes
1 answer
572 views

Inner product over finite fields

Let $F$ be a finite field, For every $c \in F$, let $X_1, X_2,..., X_9, Y_1,..., Y_9$ be independent non-zero random variables over $F$. Denote $X=(X_1,...,X_9)$, $Y=(Y_1,...,Y_9)$, also let $\...
2 votes
1 answer
220 views

Invariant characterization of isometric embeddings

$\newcommand{\Cof}{\operatorname{Cof}}$ $\newcommand{\Det}{\operatorname{Det}}$ $\newcommand{\Lam}{\operatorname{\Lambda}}$ Motivation (and the "classic" case): I am trying to find a coordinate-free ...
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0 votes
1 answer
147 views

Non-strict column diagonally dominant matrix inner product

Let $A \in \mathbb{R}^{n \times n}$ be a normalized non-strict column diagonally dominant matrix, that is: $$a_{j,j} = \sum_{i \ne j} \left|a_{i,j}\right|$$ where $0 \le a_{j,j} \le 1$ and $-1 \le ...
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2 votes
0 answers
94 views

How to calculate $\langle v,w\rangle$ based only on $\langle v,x_i\rangle$ and $\langle w,x_i\rangle$? [closed]

Let's assume $v,w, x_i \in R^n$ are unknown. Can one compute dot product $\langle v,w\rangle$ if one has just the numbers: $\langle v,x_i\rangle$ and $\langle w,x_i\rangle$ for $n$ random vectors $...
1 vote
0 answers
464 views

Complex conjugate and unitary complex conjugate

Definition: Let V be complex finite dimensional inner product space Complex Conjugate: $J$ is called complex conjugate on V iff $(i) J$ is antilinear $(ii) J^2 =I$ Definition: Anti-unitary Complex ...
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0 votes
2 answers
2k views

Different inner products for vector spaces of random variables

The inner product that appears in most books on probability is the covariance $\langle X,Y \rangle = E[XY]$ (considering that $X$ and $Y$ are zero mean real random variables). Are there other inner ...
4 votes
1 answer
336 views

Existence of $SO(n)$-isotropic inner products which are not $O(n)$-isotropic

$\newcommand{\al}{\alpha}$ Let $M_n$ be the space of $n \times n$ real matrices. Question: For which $n$, is there an inner product on $M_n$ which satisfies: $$(*) \, \, \langle Q^TXQ,Q^TYQ \...
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2 votes
0 answers
105 views

Terminology for research on distributions of inner products

Consider a set of vectors $M$ from an inner product space $V$. The ordered set of inner products of all pairs of elements in $M$ uniquely characterizes $M$ up to isomorphism. Suppose now that $V$ is ...
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