Questions tagged [inner-products]

For questions about inner products and inner product spaces, including questions about the dot product. An inner product space is a vector space equipped with an inner product. The dot product (seen in multivariable calculus and linear algebra) is a simple example of an inner product—other inner products may be seen as generalizations of the dot product.

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8 views

Prove that the derivative of a vector with constant module is perpendicular

I am told to prove that the derivative of a vector with constant module is perpendicular to the vector. Here is my approach, though I'm not sure about if it's correct or not: Let $A(t)$ be a constant ...
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Inner product formula used by numpy.dot

Dot products are pretty simple for 1- or 2-dimensional arrays, but anything beyond that is incomprehensible to me. I tried looking into numpy‘s dot function but the C code is incomprehensible to me. ...
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1answer
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How did they obtain 15-15i

This might be a silly question, but how did they get 15-15i for example 1? The operation they did didn't follow any of the definitions listed (a-d). Did they use operation a? If so, where did they get ...
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Zero inner product property [closed]

While relfecting on the properties of the inner product, I thought whether there is an easy way to show that if $\langle x,y\rangle=0\;\forall y \implies x=0$, in full generality.
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Scalar functions - Gram-Schmidt Orthogonalisation

I'm reading Chapter 11 (Normal Modes) of Classical Mechanics (5th ed.) by Berkshire and Kibble and came across this on pg. 253: The kinetic energy in terms in terms of the generalised coordinates is ...
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1answer
14 views

Show that every eigenvalue to a unitary operator has absolute value 1

Let $A$ be a unitary operator on hilbert space $H$, i.e. $$(Au|Av) = (u|v)$$ for all $u,v \in D_A.$. I'm asked to show that all eigenvalues to this unitary operator has absolute value 1. My attempt: ...
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Can we turn $\mathfrak{X}(M)$ into a Hilbert Space/ Inner product space

Given a Compact, Riemannian Manifold $(M,g)$, I'm wondering if we can induce an inner product space on $\mathfrak{X}(M)$, the set of smooth vector fields on $M$ that might have some interesting ...
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Find the point on a plane with an eqution that works per axis (not the dot product way in which is the sum of x * x, y * y, z * z).

I have a complex equation for a point and, unfortunately, I can´t simplify it. This is leading me to a complex cubic equation that I would prefer to avoid. Am currently checking the dot product ...
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61 views

Dot product terminology

How would you grammatically correctly describe the computation of the dot product of two n-dimensional vectors. Is this correct: We multiply the corresponding elements of the two vectors. Then we sum ...
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Parallel planes: lack of intuiton

I am trying to build an intuition to understand SVMs. If we have a binary set of data which is linearly separable then we want to maximize the distance between the 2 planes: $$\mathbf{wx} - b =1$$ $$\...
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1answer
33 views

Projection of $\mathbf{X}$ in the direction of $\mathbf{a}$: $\mathbf{a}^T \mathbf{X} = \sum_{j = 1}^d a_j X_j$

I am currently studying principal component analysis in statistics. PCA uses the "projection of $\mathbf{X}$ in the direction of $\mathbf{a}$": $$\mathbf{a}^T \mathbf{X} = \sum_{j = 1}^d ...
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18 views

Show an inequality using scalar product of vectors

How can I prove that: $$0<r^2<(r^2+(wl)^2)((1-w^2lc)^2+(wrc)^2)\tag1$$ $\forall c>0$ and all the other variables are bigger than zero using the scalar product of the vectors $A=(r,wl)$ and $B=...
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1answer
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Trouble deriving the value of $⟨x_i|x_i⟩$ in an infinite dimensional vector space as the Dirac Delta $δ(x_i- x_i)$

On pg. 57 of Principles of Quantum Mechanics, the author considers a vector space of "infinite dimensional vectors", conceived of as a vector space of functions defined on some closed ...
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2answers
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Taylor approximation is not optimal

My professor gave a lecture on an orthogonal polynomial based approximation and its advantage over the Taylor series expansion. And his statement was ``in weighted $L_2$ space, Taylor series expansion ...
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Unsure of steps in my simplification of work done on a surface

My attempt to simplify the integral below (found in a continuum mechanics text) employs a number of steps (e.g. involving the dot product of a scalar and a second order tensor) that I am unsure about. ...
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1answer
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Visualization of length and orthogonality under non-standard inner product

In under the standard inner product, the length of \begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix} is 1. However, under the innner product where a>0, the length becomes It is obvious to me that under ...
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1answer
45 views

Real-valued part of complex inner product

$\DeclareMathOperator{\Re}{Re}$Let $\left<f, g\right>$ be the complex inner product defined by $$\left<f, g\right>=\Re\left(\int_{-\infty}^{\infty}f^{*}(x)g(x)\,dx\right) $$ Could I say ...
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Nondegenerate dual pairings inducing isomorphisms

Given two $k$-vector spaces $A$ and $B$ and a bilinear pairing $\langle-,-\rangle: A \times B \to k$ that is non-degenerate in both entries, do we necessarily have a linear isomorphism between the two?...
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31 views

Prove unitary evolution for an ODE

Consider a nonlinear ODE of square matrix $A(t)=(a_1(t),\cdots,a_n(t))^T$ $$\mathrm{i}\,\dot A(t) = A(t) M(t)$$ with $$M(t) = A^\dagger(t) H(t) A(t)$$ where $H(t)$ and hence $M(t)$ are Hermitian ...
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121 views

Get wrong answer on $\frac{\partial \mathbf{x}^{\top} \mathbf{A} \mathbf{x}}{\partial \mathbf{x}}$ when using graph

I can use the product rule to obtain $\frac{\partial \mathbf{x}^{\top} \mathbf{A} \mathbf{x}}{\partial \mathbf{x}} = \mathbf{x}^{\top} \frac{\partial \mathbf{A} \mathbf{x}}{\partial \mathbf{x}}+(\...
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1answer
84 views

When does the Frobenius Norm equal normalized inner product?

Assume you need to design an image retrieval system where you want to find similar images to a query image $X \in \Bbb{R}^{N \times M}$ among the database images $Y_{k} \in \Bbb{R}^{N \times M}$ You ...
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2answers
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Cross, Dot Product Properties - Volume of Brillouin Zone

(See image) I'm trying to understand how the final line of this answer is reached. I know that an identity is used to get from the 3rd last line to the 2nd last line, but what properties of dot & ...
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1answer
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Why is the notation for the inner product of two columns of the Vandermonde matrix expressed this way? [closed]

I don't see how the above is a dot product for a Vandermonde matrix.
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Product rule for 2 vectors

Given 2 vector-valued functions u(t) and v(t), we have the product rule as follows. $\frac{d}{d t}[\mathbf{u}(t) \cdot \mathbf{v}(t)]=\mathbf{u}^{\prime}(t) \cdot \mathbf{v}(t)+\mathbf{u}(t) \cdot \...
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132 views

Inner products on integral lattices

I am reading Charlap's book on crystallographic groups and have trouble understanding the proof of the following proposition: Proposition. Let $\Phi$ be a subgroup of $\mathrm{GL} (n, \mathbb Z)$. ...
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1answer
53 views

Inner product in hyperbolic space

I am trying to learn about Hyperbolic Spaces. I can't find information about the inner product in hyperbolic spaces. So the paper "Multi-relational Poincaré Graph Embeddings" from Balaževic ...
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1answer
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Prove that 2 inner products are related like this when they are equal for orthogonal vectors

I'm working through Axler's "Linear Algebra Done Right", but I'm stuck on Exercise 6.B Q11: Suppose $<.,.>_1$ and $<.,.>_2$ are inner products on $V$ such that $<u,v>_1 = ...
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30 views

Prove that this dot product property does not hold

I'm asked to disprove the following statement. For vectors $\vec{x},\vec{y}\in \mathbb{R}^3:$ $(\vec{x}\cdot2\vec{y})+\vec{x}-\vec{y}=2(\vec{y}\cdot\vec{x})+\vec{x}-\vec{y}=(2\vec{x} \cdot \vec{y})+\...
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1answer
78 views

How many and which unitary transformations in $2d$

I was just wondering, probably totally trivial question, but say I have a $2d$ vector in $\mathbb{R}^2$ or $\mathbb{C}^2$ and the scalar product: Which are the possible unitary groups leaving the ...
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1answer
32 views

Sum over product of scalar products with indicator functions

Let $(h_n)_{n\in\mathbb{N}}$ be the Haar orthonormal basis. In our script we have the following equation as part of a proof for the existence of a Brownian Motion: $\sum_{n=0}^{\infty} \langle h_n , \...
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Generalization of Inner Product to Vector Spaces

For two unit vectors $\vec{u}, \vec{v}$ in an inner product space $V$, we can gauge how "similar" they are by calculating the inner-product of the two. An inner product of $1$ means they are ...
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1answer
51 views

If $\langle x,y \rangle=\|x\|\|y\|$ then $y=0$ or $x=\alpha y$

Let X be a (complex) inner product space. Proof that if for some $x,y \in X$ we have $\langle x,y \rangle=\|x\|\|y\|$ then $y=0$ or there exists complex $\alpha$ such that $x=\alpha y$ I have tried ...
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1answer
26 views

Name of matrix operation of [ A[0] dot B[0], A[1] dot B[1] ] from 2x2 matrices A, B

Please advise how this matrix operation is called, and what is the numpy operation for it. np.inner, np.dot does not create dot ...
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1answer
37 views

Confusing dot product and inner product in a weak formulation

I have been struggling with this for a while. Here, as you can see, they define the weak formulation of the Poisson equation as: $-\int_{\Omega}\nabla u\cdot\nabla v\,ds = \int_{\Omega}fv\,ds \equiv -\...
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2answers
37 views

For $x, y, z \in R^n$, show if $x \cdot y = z \cdot y$ , $x = z$

Is it right for me to do the following? $x \cdot y = z \cdot y$ $x \cdot y - z \cdot y =0$ $(x-z) \cdot y= 0$ Let y = (x-z) $(x-z) \cdot (x-z) = 0$ x = z Similarly for another question, For $x, y, z$ ...
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1answer
71 views

Difference between the definition of the dot product in infinite and finite dimensions

I know that in $\mathbf{\mathbb{R}}^n$ the definition of the dot (or scalar) product is the following: $x.y=x^{\mathrm{T}}y$, with ''T" denoting the transpose of the vector x. How does this ...
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1answer
29 views

Why does doubling a covector make the “ruler markings” closer together?

I was watching this video about how to visualize covectors as "ruler markings" or "topographic maps" with which to measure vectors. If $\alpha$ is a covector and $v$ is a vector, ...
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Hyperplane separation theorem for open sets

The hyperplane separation theorem reads Let $A$ and $B$ be two disjoint nonempty convex subsets of $\mathbb{R}^n$. Then there exist a nonzero vector $v$ and a real number $c$ such that $$ \langle x,v\...
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1answer
32 views

Orthogonal group derived using tensor notation $f(v)=v_av_bm^{ab}$, instead of inner product notation $f(v)=v^Tv$?

I wish to replicate the following proof using tensor notation. Let $v$ be a vector of $\mathbb{R}^n$. First I define this function: $$ f(v)=v^Tv\\ $$ I now ask, what are the transformations that leave ...
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1answer
22 views

Inner product at the boundary

Let $K\subset \mathbb{R}^n$ be a convex set with nonempty interior, i.e. $K^{\circ}\neq \emptyset$. Suppose there exist a nonzero vector $v$ such that $$\langle x, v\rangle\geq 0$$ for all $x\in K^{\...
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1answer
18 views

closed subspace in $W[0,1]$

Let $W[0,1]$ be the space of all continuously differentiable functions on $[0,1]$ with values in $\mathbb{C}$, with the following inner product $(f,g)_W =\int_0^1 {f(t)\overline{g(t)}+f'(t)\overline{g'...
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28 views

Self adjoint, positive definite, non elliptic operator on Hilbert space

I'm trying to find an example of a Hilbert space and a self adjoint, positive definite operator $A\in L(H)$ that is not elliptic. A suggestion given is to look at the space of square summable ...
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1answer
55 views

a dense subspace of inner product space whose codimension is 1

Is there an example of an inner product space $V$ and a dense subspace $D$ whose codimension is $1$? In other words, there exists $z\in V$ such that every element of $V$ can be written in the form $e+\...
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About the “dot product” in the material derivative

Recently the material derivative was mentioned in my fluid dynamics class. The definition given is $$\operatorname{MD}=\partial_t +\mathbf{u}~\boldsymbol{\cdotp}\nabla$$ Where of course $\nabla$ is ...
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1answer
96 views

An upper bound of product of two inner products

The question is, Let $A \in M_{n \times n}(\Bbb C)$ be a self-adjoint matrix. Arrange the eigenvalues of $A$ as $0 < \lambda_1 \le \lambda_2 \le \cdots \le \lambda_n. $ Show that $ \langle Ax, x \...
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1answer
21 views

Adjoint operator on a Hilbert space

I am working on this problem: Consider the space $H=\{f:\mathbb{R}\to\mathbb{C} : \int_{-\infty}^{+\infty} e^{-|x|} |f(x)|^2 dx <\infty\}$, this is a Hilbert space with the inner product $<f,g &...
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1answer
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self-adjoint operator is bounded in Hilbert space [closed]

$X$ is a Hilbert space. $A$ is a linear and defined everywhere on $H$. $A$ satisfies $$<x,Ay>=<Ax,y>$$ for all $x,y\in H$, then $A$ is bounded. $<,>$ denote the inner product
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1answer
23 views

Symmetric Bilinear Form/ Hermitian Form unique Matrix representation

Given $\mathbb{K} = \mathbb{R} $ or $ \mathbb{C}$ I have $\varphi:\mathbb{K^n}\times\mathbb{K^n}\rightarrow \mathbb{K}$ a symmetric Bilinear Form or complex Hermitian Form/Symmetric sesquilinear form ...
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1answer
57 views

Why does calculating the return on a portfolio differ when calculated at a stock level rather than at a portfolio level?

I have a portfolio of three assets A, B and C, each with a beginning value of 100. I trade the portfolio over two days. On day 1, asset A returns 50, B returns 20, and C returns -10. On day two, A ...
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1answer
45 views

Proving something is an inner product

I am faced with the following question: Prove that this is an inner product on V. I understand to show something is an inner product I must verify the following conditions: $\langle\,f,g\rangle = \...

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