Questions tagged [inner-product]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
-3
votes
1answer
60 views

Derivative of dot product [closed]

The dot product of two distributions u(s) and v(s) might be written as $u(s)^{\intercal} \, v(s)$. What would be a closed form for the derivative $\frac{\partial}{\partial s} \left( u(s)^{\intercal} \,...
4
votes
0answers
53 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
0answers
29 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 ...
2
votes
0answers
133 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
1answer
55 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
1answer
121 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 (...
2
votes
1answer
77 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
1answer
141 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 ...
5
votes
2answers
433 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
1answer
278 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 ...
1
vote
1answer
490 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 ...
17
votes
2answers
435 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 ...
2
votes
1answer
140 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 $$ \...
2
votes
2answers
191 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 $\mathbb{R}$. ...
1
vote
1answer
263 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 ...
6
votes
1answer
168 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 ...
4
votes
1answer
351 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 ...
2
votes
0answers
81 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
0answers
604 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
0answers
120 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 ...
13
votes
1answer
538 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
0answers
114 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}...
0
votes
0answers
162 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
0answers
87 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, ...
1
vote
0answers
32 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 ...
9
votes
1answer
478 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
1answer
213 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 ...
0
votes
1answer
138 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 ...
2
votes
0answers
91 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
0answers
366 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 ...
0
votes
2answers
730 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
1answer
237 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 \...
2
votes
0answers
75 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 ...
13
votes
1answer
659 views

Maximum number of vectors in a hypercube satisfying given conditions

$\mathcal{C}$ is a collection of binary vectors of length $n$, i.e. $\mathcal{C}\subseteq\{0,1\}^n$. For arbitrary $x,y,z\in\mathcal{C}$ and $x\neq z$, $y\neq z$, there always holds that the Euclidean ...
1
vote
2answers
150 views

Generalised “projection” of a metric space

Assume we have $n$ points $p_0\ldots p_{n-1}$ which form a discrete metric space $V$ with metric $d$. Can we define a function $f:V\rightarrow \mathbb{R}$ with $f(p_0) = 0$, $f(p_1) = d(p_0,p_1)$ and $...
0
votes
2answers
113 views

Inner Product of Given Sum Positive Sequence

Let $$A = \Big\{(a_1,a_2,\dots)\ \Big|\ a_i\ge 0, \sum_{i=1}^\infty a_i=1\Big\},$$ $$v(x)=\sup\left(\bigg\{\sum_{i=1}^\infty a_ib_i\ \bigg|\ (a_i)_{i=1}^\infty,\, (b_i)_{i=1}^\infty \in A,\,\sup\...
4
votes
1answer
304 views

A Hilbert-space completion of a Hilbert $ C^{*} $-module over a separable $ C^{*} $-algebra

Let $ B $ be a separable $ C^{*} $-algebra and $ \mathcal{E} $ a Hilbert $ B $-module. We know that $ B $ has a faithful state $ \phi $. Using $ \phi $, we can construct a $ \mathbb{C} $-valued pre-...
0
votes
1answer
105 views

Is the Hodge Map Unitary?

Let $(V,< \cdot, \cdot >)$ be an inner product space over a field ${K}$. As usual, we can extend $< \cdot, \cdot >$ to a mapping on the exterior algebra of $V$ using the usual matrix ...
15
votes
1answer
2k views

A property that forces the NORM to be induced by an INNER PRODUCT

Let $(E, \|\cdot\|)$ be a real normed vector space such that for any $a,b\in E$, $$ \|x +y\|^2 + \|x-y\|^2 \geq 4 \|x\|\cdot \|y\| $$ I want to show that the norm is induced by an inner product. Any ...
3
votes
1answer
177 views

Classifying all Equivariant Bilinear Forms on a Finite-Dimensional Module

Given a finite dimensional (real) vector space $V$, and two non-degenerate bilinear forms $(\cdot,\cdot)_1$ and $(\cdot,\cdot)_2$, one can use a basic linear algebra argument to show that there exists ...
8
votes
2answers
1k views

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 ...
9
votes
1answer
1k views

What fields can be used for an inner product space?

Math people: 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 ...
1
vote
0answers
144 views

Functional Analysis Generalizations: indeterminated inner product and functions over manifolds

There are books or articles that deal with generalizations of functional analysis in the sense that the inner product need not be positive-definite or that works with functions over manifolds?
1
vote
0answers
384 views

How does the Schmidt decomposition generalize to tensor products of several finite-dimensional systems?

Let $n,d_1,\ldots,d_n > 1$ be integers, and $V_1, \ldots, V_n$ be inner product spaces over $\mathbb C$, having dimensions $d_1, \ldots, d_n$ respectively. We consider the ways in which we may ...
16
votes
2answers
3k views

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 ...
11
votes
1answer
613 views

Unitary representations of Quantum Groups

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and let $U_q(\mathfrak{g})$ be some incarnation of the quantized universal enveloping algebra of $\mathfrak{g}$; here I am ...
3
votes
2answers
383 views

Inner product spaces, Siegel's theorem and lattices: book suggestion

Background: I am a theoretical computer scientist (PhD candidate) and have done graduate level courses in Algebra. I want to understand the following theorem from the book "Symmetric Bilinear Forms" ...
18
votes
7answers
7k views

What is a complex inner product space “really”?

This is an extended re-post of a question that I have asked on MSE not a long time ago. But anyway, it seems more appropriate for MO. To begin with, in a real inner product space we have a geometric ...
5
votes
3answers
821 views

adjoint of multiplication operator in a commutative algebra

Dan Popescu asked me the following question, and since I'm not an expert I'm throwing his question on MO. Suppose that $A$ is a finite-dimensional vector space over an ordered field $k$ with $\...
18
votes
2answers
6k views

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 ...