Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [inner-product-space]

An inner product space is a vector space equipped with an inner product. The inner product is a generalization of the “dot” product often used in vector calculus.

3
votes
4answers
137 views

What is relationship between scalar and vector product?

I am intrested in relationship between scalar and vector product in $\mathbb{R}^3$; I am going to give definitions which I will use in my question. Scalar product - function $\cdot:\mathbb{R}^3 \...
1
vote
0answers
27 views

Computing the derivative of an inner product

I want to differentiate (1) wrt the vector $x \in \mathbb{R}^n$ where $w(x)$ is a zero-one diagonal indicator matrix \begin{align} \frac{1}{2} g(x)^T w(x) g(x) &&&& (1) \end{align} ...
5
votes
2answers
49 views

Is it possible to recognize when an endomorphism of a finite dimensional vector space is unitary for some choice of inner product?

Let $V$ a finite dimensional vector space over $\mathbb{C}$. Let $T\in GL(V)$. Are there reasonable criteria for recognizing whether or not there is some inner product on $V$ w.r.t. to which $T$ is ...
2
votes
1answer
27 views

Show that this is an inner product

Let's define $$(f,g)=\int_{\mathbb{R}} \frac{f(x)\bar{g}(x)}{1+x^2}dx$$ $\forall f,g\in X=\{h:\mathbb{R}\rightarrow\mathbb{C}:$ $h$ is Lebesgue-measurable and bounded over $\mathbb{R}$} I have to ...
0
votes
2answers
30 views

Proof of one of the dot product theorem

This is a statement I came up on my own and it makes intuitive sense but I'm not sure how to go about proving it. (Also I'm sure this exists already but couldn't find it online for odd reasons.) ...
1
vote
1answer
33 views

Conjugate symmetry of an inner product

I want to prove the following: $\langle A,B \rangle = \overline{\langle B,A \rangle}$ where $\langle A,B \rangle := tr(AB^{*})\,\, and \,\, A,B \in \mathbb{C}^{n \times n} $ Note: The bar ...
2
votes
2answers
27 views

If $A = A^*$ then $ \lambda_{min} \leq\frac{\langle Av,v \rangle}{\langle v,v \rangle}\leq\lambda_{max} $

I'm leaning linear algebra and new to it. I have trouble with this problem and actually, I don't know what to do! any help or hint would be appreciated. for the linear operator $A\in \ell(V)$ wich $...
-1
votes
2answers
36 views

The Order of Orthogonality [closed]

I would like to show that $B\subset A$ implies $A^{\bot}\subset B^{\bot}$. Note the meaning behind this: The bigger a subset, the smaller its orthogonal should be. Let $x$ be in the complement of A. ...
2
votes
1answer
18 views

What is the expectation of the product of a normalized (complex) gaussian vector and its hermitian transpose?

I have a complex Gaussian vector $\mathbf{x} \triangleq (x_1; x_2; ...; x_n) \in \mathbb{C}^{n \times 1}$, where $x_i \sim \mathbb{CN}(0,1)$ i.i.d., and $\mathbf{y} \triangleq (y_1;y_2;..;y_n)\in \...
0
votes
0answers
26 views

Proof: Matrix has just non-negative real eigenvalues

$\textbf{So my given problem is:}$ $Let\,\, A \in \mathbb{C}^{n \,\times \,n}\,\, be\,\, such\,\, that\,\,\\ \forall x\in \mathbb{C}^n : \langle\,Ax,x\rangle \geq 0 \\ where \,\, \langle\,\cdot,\cdot\...
2
votes
2answers
29 views

Orthogonal matrices only defined for standard inner product?

$\newcommand{\tp}[1]{#1^\mathrm{T}} \newcommand{\Id}{\mathrm{Id}} \newcommand{\n}{\{1,\ldots,n\}} \newcommand{\siff}{\quad\Leftrightarrow\quad} \newcommand{\ijth}[2][\tp{Q}Q]{[#1]_{#2}} \newcommand{\K}...
0
votes
2answers
26 views

Tensor Dot Product of two tensors of arbitrary order

I am currently working on implementing the inner(scalar or dot) product of two tensors of arbitrary order. As far as i understand, you need to make sure, that the last dimension of the first tensor $\...
0
votes
0answers
19 views

Linear Algebra - Positive-Definiteness in Vectors?

I was reading up on the inner product over at this Wikipedia page, and I noticed, in the given definition, the use of the term "positive-definiteness". Now, from what I know, this is terminology one ...
0
votes
1answer
22 views

Norms and Parallelogram Identity

I have recently started learning about Norms and Inner Products. I have came across the idea that for an inner product to introduce a norm the parallelogram Identity must be true. The proof my books ...
1
vote
1answer
21 views

L2 and Scalar Product

I struggle to show that for $\, f,g\in L^2(X,A,\lambda)$ this is a scalar product: $ \langle f,g\rangle := \int fg \, d\lambda $ It wasn't hard to show that the length is positive, but I struggle ...
0
votes
1answer
30 views

Prove that $(f, g) = \int_{-1}^{1} f(t)g(t) \,\mathrm dt$ defines an inner product.

Let $V$ be the vector space of all continuous functions $f\colon [-1,1]\to\Bbb R$. Prove that $(f, g) = \int_{-1}^{1} f(t)g(t) \,\mathrm dt$ defines an inner product. I'm fairly new to inner products ...
1
vote
0answers
26 views

Why two vectors' covariance is the dot product of these two vectors

I am trying to understand the OLS property that SST(Total sum of squares) = SSE (explained sum of squares) + SSR (residual sum of squares). One of the steps is to prove that sample covariance of ...
1
vote
0answers
43 views

Proving that a function f is a kernel.

The theory states that $$f(x,y)\\ \text{with} \ x\ \text{and}\ y \in R^n$$ in order to be a valid kernel, beside being symmetric, has to be an inner product in a suitable space. Is this latter ...
1
vote
1answer
27 views

Determine vector c, which is collinear vector of vector $a+b$

Determine vector c, which is collinear vector of vector $a+b$, if $ab=5$, $cb=18$ and $|b|=2$. I tried with $c=n(a+b)$. $9= |c|*cos(\alpha)$...$|c|=\sqrt{(n^2(a+b)(a+b))}= n \sqrt{aa+14}$ Then $9= ...
0
votes
2answers
27 views

Scalar product of complex valued square integrable functions

So I was told that if $p,q \in \mathbb{L}^2 _\mathbb{C} [a,b]$ which is the set of all square integrable functions on the interval $[a,b]$ then: $$(p,q) = \int^b_a{dx\ p^*(x)q(x)}$$ is a scalar ...
1
vote
1answer
33 views

Is $\mathbb{R}^n$ always an inner product space?

This may be a silly question but every two norms on $\mathbb{R}^n$ are equivalent and $\Vert\cdot\Vert_2$ comes from the usual dot product so $(\mathbb{R}^n,\Vert\cdot\Vert_2)$ is (at least) an inner ...
1
vote
1answer
19 views

Finding the norm of a matrix given an orthonormal basis

So I am working on practice problems out of a textbook I am working through over break. One of the problems asks to show that given a complex $n\times n$ matrix $X$ and any orthonormal basis $\{u_1,...
2
votes
1answer
36 views

Prove that Gramian matrix is Invertible iff $(v_1,…,v_k) $ is linearly independent

Prove that Gramian matrix is Invertible iff $(v_1,...,v_k) $ is linearly independent $G=G(v_1,...,v_k) = [\left\langle v_i,v_j\right\rangle ]_{i,j=1}^k $ I have great idea to calculate $\det G$ and ...
1
vote
1answer
40 views

The value of k such that two vectors are orthogonal in integral product spaces

For $f,g \in C[0,1]$ we define inner product space $\langle f,g \rangle =\int_0^1 f(x)g(x) dx$. Find the value of $k$ such that $f(x)=\sin(kx)$ and $g(x)=\cos(7x)$ are orthogonal in this inner product ...
1
vote
1answer
13 views

Equivalent Conditions of Nondegenerate Bilinear Forms and the Gram Matrix

One can often find the following theorem describing equivalent conditions for non degenerate bilinear forms. $\textbf{Theorem 1}$: Let $V$ be a vector space over the field $\mathbb{F}$ equipped with ...
1
vote
0answers
13 views

Total subsets in incomplete inner product spaces [duplicate]

I'm reading Induced Representations of locally compact groups by Kaniuth and Taylor and I don't understand how total subsets work (in particular in Lemma 2.24). I know that a total subset of an inner ...
0
votes
2answers
48 views

How to undo a matrix-vector multiplication

I have an iterative algorithm that computes a matrix-vector multiplication such as: $$ b = Av $$ I know the vector b (which is the result of the algorithm) and the vector v. Is there a way to get ...
0
votes
1answer
30 views

Verifying an inner product

Let $E$ be the linear space of all continuous complex-valued functions defined on the interval $(a,b)$ of the real line. Consider the inner product in $E$ $\langle f,g \rangle = \int_a^{b} f(x) \...
2
votes
1answer
36 views

Inner product with orthogonal complement

Let $\mathbb{R^3}$ be equipped with the inner product $<,>$ defined by setting $$<\mathbf{u},\mathbf{v}>=2u_1v_1+5u_2v_2+3u_3v_3$$ for any pair of vectors $\mathbf{u}=(u_1,u_2,u_3)$ ...
2
votes
1answer
43 views

Maximize product of a vector with two vectors

Say we're in a (complex) Hilbert space $H$ where we're given some two elements $a,b\in H$ of norm 1. The question is to find an element $\psi\in H$ (of norm 1) that maximizes $|\langle \psi,a\rangle \...
2
votes
3answers
88 views

Proving that $\langle \textbf {a,b} \rangle = \|\textbf {a}\| \|\textbf {b}\| \cos \theta$.

How to show that $\langle \textbf {a,b} \rangle = \|\textbf {a}\| \|\textbf {b}\| \cos \theta$ where $\langle \cdot , \cdot \rangle$ denotes the usual real inner product or dot product on $\Bbb R^2$, $...
2
votes
1answer
44 views

Cauchy Schwarz inequality for $\langle f,g\rangle = \int_0^1f(x)g(x)\mathrm{d}x$ [duplicate]

For the vector space of continuous functions on $[0,1]$ Define the inner product as $$\langle f,g\rangle = \int_0^1f(x)g(x)\mathrm{d}x$$ Please help me to prove the Cauchy Schwarz inequality for this ...
0
votes
1answer
18 views

Resulting distribution from inner product between vector which entries are normally distributed and another vector

I need to understand the following part of lectures notes (below). How did we obtain distribution of inner product outcome (in red box)? It is probably some obvious rule that I do not know.. Thank ...
0
votes
2answers
28 views

Why convergence of these two series is equivalent?

In my notes I have the following: For $s\geq 0$, define the Sobolev Space $H_s$ by: $$H_s := \left\{f\in L^2([0, 2\pi], \mathbb{C})\,\, : \,\, \sum_{k=-\infty}^\infty |k|^{2s}|\hat{f_k}|^2 <\...
5
votes
1answer
121 views

When is a topological vector space “inner product-able”?

This is a follow-up to my question here. A topological vector space is normable, i.e. its topology is induced by some norm on the vector space, if and only if it is Hausdorff and the $0$ vector has a ...
3
votes
1answer
77 views

Is every normable topological vector space “inner productable”?

Not every norm on a vector space is induced by an inner product on that vector space. But suppose hat $X$ is a topological vector space that is normable, i.e. its topology is induced by some norm on ...
0
votes
0answers
24 views

Basis of an Orthogonal Complement

Let $V$ be the vector space of all $2$ by $2$ matrices. Let $<M_1, M_2>$ $=$ $tr(M_1^TM_2)$ be an inner product defined on $V$. Let $A$ $=$ $\begin{pmatrix}1&1\\ 1&0\end{pmatrix}$ be one ...
0
votes
0answers
36 views

Proving an inner product

Let $V$ be the vector space of all $2$ by $2$ matrices. Define $\langle A, B\rangle = \mathrm{tr}(A^TDB)$, where $D = \begin{pmatrix}7&1\\ 1&1\end{pmatrix}$. Prove that $<A, B>$ defines ...
0
votes
2answers
22 views

Element not orthogonal to any other non zero element

Consider an inner product space. Does there(always) exist a non zero element which is not orthogonal to any other non zero element? In other way, can I always find a non zero element which is ...
8
votes
2answers
65 views

Show that the $L^{p}$ norm $\|f\|_{L^{p}} := \big( \int^{b}_{a} |f(x)|^p\big)^{1/p}$ is not induced by a scalar product for $p \neq 2$.

On $X = C^0\big([a,b]\big)$, for any $p \in \mathbb{R}$, $p>1$, we define the $L^p$ norm by, $$\|f\|_{L^{p}}:=\big(\int^{b}_{a}|f(x)|^{p}dx \big)^{1/p}.$$ Show that for $p\neq 2$, this ...
0
votes
1answer
30 views

Cross Products/Determinants/Matrix Multiplication Under Arbitrary Inner Products

This video explains that the cross product and the determinant involve the dot product under the hood. This video explains that the most fundamental, entry-wise perspective of matrix multiplication ...
3
votes
3answers
81 views

In $\mathbb{R}^n$, is the dot product the only inner product?

Just what the title says. I'm reading, from various resources, that the inner product is a generalization of the dot product. However, the only example of inner product I can find, is still the dot ...
1
vote
0answers
29 views

Distance between two polynomials (inner product)

I don't know how I've gotten this question wrong. I have to compute the distance between: $f(t) = 2t + 3$ and $g(t) = 3t^2 -1$ Their inner product is defined as $\int_{0}^{1}f(t)g(t)dt$ So I ...
0
votes
0answers
23 views

Calculate dot product without the use of angles

In a book called Introduction to tensor analysis and the calculus of moving surfaces equation (2.6) gives a formula to calculate the dot product between two vectors in terms of length alone. That ...
1
vote
1answer
52 views

Non total orthonormal set in a non Hilbert inner product space

Suppose there exist a subset $M$ of an inner product space $X$, and the orthogonal complement of $M $ is the zero vector. If $X $ is a Hilbert Space then the span of $M $ will be dense in $X $, but ...
1
vote
2answers
38 views

Showing that $Y = \cup_{n=1}^\infty Y_n$ is dense over the Hilbert $H$.

Source of question : Trying to prove that if the Hilbert $H$ has an orthonormal basis, then it is separable. Elaboration : Let $H$ be a Hilbert space and $\{e_n : n \in \mathbb N\}$ an orthonormal ...
2
votes
2answers
27 views

$v \subseteq H \implies V^\bot$ is a closed subspace of the Hilbert space $H$

Exercise : Show that if $H$ is a Hilbert space and $V \subseteq H$, then $V^\bot$ is a closed subspace of $H$. Attempt : I thought of two possible approaches. One would be a classic one, getting ...
2
votes
1answer
42 views

Inner product of two vectors in complex vector space in index notation

Suppose $\vec{u}, \vec{v} \in \left(V, \mathbb{C}^{n}\right)$: by construction $$ \begin{split} \vec{u} &= \sum_{i=1}^{n}u_{i}\vec{e}_{i}\\ \vec{v} &= \sum_{i=1}^{n}v_{i}\vec{e}_{i} \end{...
1
vote
1answer
74 views

Inner Product and Operators: Exercise with vectors with three indices

I have the following question: I started off as follows: Let $v\in \ell(\mathbb{N}, \mathbb{C})$, and $w,x$ defined as above. Then: \begin{align} \langle w, x\rangle_J &= \sum_{(i,j,k)\in J}w_{[i,...
0
votes
3answers
41 views

Every inner product space is a metric space.

Show that every inner product space is a metric space. To show this should I set the distance metric as $d(x,y) = <x-y,x-y>$, then show properties of being metric space such as d(x,y) = d(y,x) ...