Questions tagged [inner-product]
The inner-product tag has no usage guidance.
86
questions
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 $$(\cdot)^T (\...
34
votes
5
answers
13k
views
Linearity of the inner product using the parallelogram law
A norm on a vector space comes from an inner product if and only if it satisfies the parallelogram law. Given such a norm, one can reconstruct the inner product via the formula:
$2\langle u,v\rangle ...
32
votes
4
answers
9k
views
Definition of inner product for vector spaces over arbitrary fields
Is there a canonical definition of the concept of inner products for vector spaces over arbitrary fields, i.e. other fields than $\mathbb R$ or $\mathbb C$?
22
votes
7
answers
9k
views
What is a complex inner product space "really"?
This is an extended re-post of a question that I have asked on MSE not a long time ago. But anyway, it seems more appropriate for MO.
To begin with, in a real inner product space we have a geometric ...
20
votes
3
answers
7k
views
Why do inner products require conjugation?
For Hermitian matrices and operators, the most "natural" inner product is $f^H \cdot g$ or $\int f^* g\; dx$. A similar situation holds interpreting Fourier transforms as the inner product of ...
18
votes
2
answers
4k
views
Minimum off-diagonal elements of a matrix with fixed eigenvalues
I am an engineer working in radar research. I came accross a problem on which I cannot seem to find literature. I can ask it in two different ways. Perhaps depending on the reader, the alternative ...
17
votes
2
answers
506
views
On a special type of normed linear spaces
Let $(V,\|.\|)$ be a normed linear space such that for every group $(G,*)$, every function $f:G \to V$ satisfying
$$
\|f(x*y)\|\ge \|f(x)+f(y)\|,\qquad\forall x,y\in G,\tag{Z}
$$ is a group ...
16
votes
1
answer
3k
views
A property that forces the NORM to be induced by an INNER PRODUCT
Let $(E, \|\cdot\|)$ be a real normed vector space such that for any $a,b\in E$,
$$ \|x +y\|^2 + \|x-y\|^2 \geq 4 \|x\|\cdot \|y\| $$
I want to show that the norm is induced by an inner product. Any ...
13
votes
1
answer
679
views
Vectors that are almost orthogonal on average: lower bounds on dimension?
Let $v_1,\dotsc,v_k \in \mathbb{R}^d$ be unit-length vectors such that
$$\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 $...
13
votes
1
answer
747
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 ...
11
votes
1
answer
949
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, \...
11
votes
1
answer
710
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 ...
10
votes
2
answers
2k
views
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, ...
9
votes
1
answer
624
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 $\...
8
votes
2
answers
2k
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 ...
8
votes
1
answer
430
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 ...
7
votes
2
answers
1k
views
Reference for the existence of a Shapovalov-type form on the tensor product of integrable modules
Shapovalov and Jantzen showed us how to construct a nice inner product on finite dimensional representations of a semi-simple Lie algebra, by simply giving the highest weight vector inner product 1 ...
6
votes
2
answers
234
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 ...
6
votes
2
answers
295
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 ...
6
votes
1
answer
274
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 ...
6
votes
0
answers
1k
views
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)...
5
votes
4
answers
1k
views
An inner product that makes the R-matrix unitary
So, if you talk to the right people, they will tell you that the braiding of the category of representations of a quantum group are not unitary and that one can fix this by taking a different commutor ...
5
votes
3
answers
1k
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 $\...
5
votes
2
answers
338
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_\...
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 \...
5
votes
3
answers
1k
views
Proof that domains of positivity of symmetric nondegenerate bilinear forms are self-dual cones?
Max Koecher (for example, in The Minnesota Notes on Jordan Algebras and Their Applications; new edition: Springer Lecture Notes in Mathematics, number 1710, 1999), defined a domain of positivity for a ...
5
votes
2
answers
472
views
Can finitely many hermitian (positive-semidefinite) operators always be embedded into a small dimensional space preserving inner product?
Given $n$ hermitian (positive-semidefinite) operators $Q_1,\cdots,Q_n$ in finite dimensional Hilbert space (the dimension can be very large), is there a mapping $\phi$ maps $Q_i$ to $P_i$, which ...
4
votes
1
answer
833
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 ...
4
votes
1
answer
369
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:
$$(*) \, \, \langle Q^TXQ,Q^TYQ \...
4
votes
1
answer
371
views
A Hilbert-space completion of a Hilbert $ C^{*} $-module over a separable $ C^{*} $-algebra
Let $ B $ be a separable $ C^{*} $-algebra and $ \mathcal{E} $ a Hilbert $ B $-module. We know that $ B $ has a faithful state $ \phi $. Using $ \phi $, we can construct a $ \mathbb{C} $-valued pre-...
4
votes
1
answer
402
views
On a vanishing integral inner product
Let $G(z)$ be an $n\times m$ rational matrix-valued function of full column rank on the unit circle. Further, let $P(z)$ be an $m\times m$ rational matrix-valued function positive definite on the unit ...
4
votes
0
answers
170
views
What is the exponential map from the Lie algebra $\mathfrak{sl}(2,\mathbb{C})\ltimes_\textrm{ad}\mathfrak{sl}(2,\mathbb{C})$ to its Lie group?
$\DeclareMathOperator\ad{ad}\DeclareMathOperator\Ad{Ad}\DeclareMathOperator\Exp{Exp}\DeclareMathOperator\SL{SL}\DeclareMathOperator\sl{\mathfrak{sl}}$Let $G:=\SL(2, C) \ltimes_{\Ad} \SL(2,C)$, where $\...
4
votes
0
answers
129
views
Independent inner functions on the unit disk
This problem cropped up in a paper that I am writing and I have thought about it for too long to no avail: let $\mathbb{D}$ be the open unit disk in the complex plane and suppose $\varphi:\mathbb{D}\...
4
votes
0
answers
120
views
Triple Petersson Inner Products With Theta Functions
Our current work requires us to bound triple products of the form
$$\langle \Im(*)^{\frac{3}{2}}|\theta^3|^2,\mu_j \rangle,$$
where $\langle \cdot, \cdot \rangle$
is the Petersson inner product, ...
3
votes
1
answer
462
views
abc-conjecture and positive definite kernels, again?
One formulation of the abc-conjecture is:
$$\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(...
3
votes
2
answers
496
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" ...
3
votes
1
answer
891
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 ...
3
votes
1
answer
240
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 ...
3
votes
1
answer
428
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 ...
3
votes
0
answers
104
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 ...
3
votes
0
answers
125
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>\...
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=\...
3
votes
0
answers
192
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 $$\hat a=(a_n,a_{n-1},\dots,a_1)\in(\mathcal I_{m_1}+\sqrt{-1}\cdot\mathcal I_{m_2}...
2
votes
2
answers
245
views
An inner product on the vector space $\mathbb{R}[x_1,\cdots,x_n]_m$
For any given integers $m,n\geq1$, let $\mathbb{R}[x_1,\cdots,x_n]_m$ be the vector space of homogeneous polynomials of degree $m$ in $x_1,\cdots,x_n$ over the field of real numbers $\mathbb{R}$. ...
2
votes
1
answer
350
views
Inner product over finite field
sorry for informals but is my first post.
In Coding theory (exactly in Coding theory a first course - San ling and Chaoping) found this definition:
$\langle , \rangle :\mathbb{F}_q^n\times\mathbb{F}_q^...
2
votes
1
answer
135
views
Largest number of two series vectors with negative dot product
Assume $\{x_{i}\}_{i=1}^{m}$, $\{w_{i}\}_{i=1}^{m}$ are two sets of vectors in $\mathbb{R}^{n}$. And we have that $ x_{i}\cdot w_{j} < 0$ for $i \neq j$ and $x_{i}\cdot w_{i} > 0$ for all $i$. I ...
2
votes
1
answer
381
views
Mean squared absolute value of inner product of unit vectors
Given a nonempty finite subset $S$ of the unit sphere of $d$-dimensional complex Hilbert space, let $\lambda(S) = \frac{1}{\lvert S \rvert^2} \sum_{x,y \in S} \lvert \langle x,y \rangle \rvert^2$ be ...
2
votes
1
answer
173
views
Controlling angles between vectors using sum of subvector angles?
This is a technical question coming out of my research.
Let $\angle(\cdot, \cdot)$ be the angle ($\in [0, \pi]$) between vectors. Consider two vectors $u, v$ in $\mathbb R^3$. Is it true that
$$
\...
2
votes
1
answer
329
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 \...
2
votes
1
answer
228
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 ...