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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Is there a equivalent norm on $L^p$ induced by inner product?

Suppose $L^p[a,b]$ is the normed space with the usual norm $\|f\|_p=(\int_a^b|f(x)|^p\mathrm{d}x)^{1/p}$. By the parallelogram equality, we know the norm is induced by an inner product iff $p=2$. ...
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Properties of conformal maps

A linear map $T:X\to Y$ is said to be conformal when it preserves orthogonality; $ \forall x,\tilde x\in X, \langle x,{\tilde x}\rangle =0\iff\langle{Tx},{T\tilde x}\rangle=0.$ Show that if $T$ is ...
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A matrix with inner product that respects norm is a reflection matrix

Let $\langle \cdot | \cdot\rangle$ be the standard inner product over $\mathbb{R}^{2}$ and let $A \in M_{2}(\mathbb{R})$ be a matrix such that $\|Av\| = \|v\|$ for every $v \in \mathbb{R^{2}}$ and ...
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The norm and the Fourier coefficient if the inner product $\langle f(x), f(x)\rangle$ is negative.

For a real function $f_m(x)$ orthogonal with respect to $w(x)$ where $w(x)>0$. We have inner product $$ \langle f_m,f_n\rangle_{w}=\int_{x_0}^{x_0 + T}{w(x)f_m(x)f_n(x)dx} $$ The norm is $||f_m||_{...
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What is $\langle x, \varphi \rangle \overline{\langle x, \varphi \rangle}$ equal to?

I'm working on the following question: Let $X$ be a Hilbert space with dim($X$)≥2 and let $T\in B(X)$ be of the form $$Tx=\langle x,\varphi\rangle\psi$$ Show that $||T||=||\varphi|| \hspace{0.3em}||\...
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Doubt from the proof that the sequence space $l^2$ is a Hilbert space from Kreszig book

This is the proof given in Kreszig book. 3.1-6 Hilbert sequence space $l^2$. The space $l^2$ (cf. 2.2-3) is a Hilbert space with inner product defined by $$ \langle x, y\rangle=\sum_{j=1}^\infty \...
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Prove that $T = cU$, where c is a scalar and $U$ is unitary.

In $\mathbb{C}^n$, with inner product $\langle z,w \rangle = \sum_{j} z_j\overline{w}_j$. Let $T$ a linear operator such that $\langle T(z), T(w) \rangle = 0$, if $\langle z,w \rangle = 0$. Prove ...
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Find inner product

I need to find an inner product in $\mathbb{C}^2$ such that $\{(1,i),(-1,i)\}$ is orthonormal. I tried the following: I know that $\langle(1,i),(1,i)\rangle=\langle(-1,i),(-1,i)\rangle=1$ and $\langle(...
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Given that the two vectors 8a -b and 4a +3b are perpendicular and that |a |= 2|b |, determine the angle between a and b. [closed]

I understand you have to use the dot product rule, but I'm not sure where to take it from there.
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Scalar Triple Product proof

I wanted to prove that if we change order of vectors involved in a scalar triple product in a cyclic fashion , then the product remain same . I want an elegant proof of it involving simple algebra of ...
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What does it mean if the inner product of two complex vectors is real and $\gt $ 0?

Does it mean these two vectors are linearly dependent or is there any other property can be derived from it?
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About the inner product and the gradient

It might be very basic, but I am curious about the calculation of $(V\nabla V)\cdot n_{\rm out}$ where $V$ is defined on $\Omega$ and $n_{\rm out}$ is unit normal tangent vector on $\partial \Omega$. ...
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Proving a formula about inner product of a vector and its projection onto a orthogonal complement of the column space

We have $h_1,...,h_{i},...,h_K \in \mathbb{C}^N, N\gt K$ and they are linearly independent. Then we define $v \triangleq \prod^{\bot}_{[h1,...,h_{i-1}\ ,\ h_{i+1} \ ,...,h_K]}h_i $, where $\prod^{\bot}...
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Proving $\sup\limits_{\theta,\psi}\{||e^{i\theta}x+e^{i\psi}y||^2:\theta,\psi\in \Bbb R\}=||x||^2+||y||^2+2 Re<x,y>$ where $x,y \in \mathbb{C}^n$

Q. Let $x,y \in \mathbb{C}^n$. Consider $f(x,y)=\sup\limits_{\theta,\psi}\{\lVert e^{i\theta}x+e^{i\psi}y\rVert^2\colon \theta,\psi\in \Bbb R\}.$ Which of the following is/are correct? $f(x,y)\leq\...
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Proving a linear transformation that preserves the inner product is an isometry

I am currently working on a problem to prove the following statement: Suppose $T: \mathbb{R}^n \to \mathbb{R}^n$ is a linear transformation. Prove that $T$ is an isometry if and only if $T(v) \cdot T(...
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How to find a basis for the kernel of T that is orthogonal with respect to an inner product?

enter image description here What is the question asking in the second part?
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Exercise 6.A.17 in "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. I am worried if my solution is ok.

I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. 6.A.17 Prove or disprove: there is an inner product on $\mathbb{R}^2$ such that the associated norm is given by $$||(x,...
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Parallel vector and dot product

I'm trying to show that $\|u\cdot v\|= \|u\| \, \|v\| \iff u = \alpha v, \quad \alpha\in \mathbb{R}$. $(\Leftarrow)\;$ Note that $\|\alpha v\cdot v\| = |\alpha| \, \| v\cdot v\| = |\alpha| \, \| (\...
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Convergence of a sub-sequence

I am having a doubt about the convergence of a sub-sequence as follows. Knowing that sequences $x(k),y(k) \in \mathbb{R}^n$ converge to $x^*, y^*$ respectively as $k$ goes to $0$ ($k \in \mathbb{R}^+$...
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What does the dot product of a tensor and a vector represent?

The dot product (or inner product) of a tensor T and a vector a produces a vector b = T . a: $$ b_i = T_{ij}a_j = \begin{pmatrix} T_{11} a_1 + T_{12} a_2 + T_{13} a_3\\ T_{21} a_1 + T_{22} a_2 + T_{23}...
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Finding Best approximation in space with standard inner product

I'm new to functional analysis. I do not know how to solve the below problem. I know that for g to be the best approximation of f, it has to satisfy the condition that (f−g)⊥G. But I could not apply ...
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In a product space $E$, and subspace $F \subset E$, $\|u-v\| \geq \|v\|, \forall v \in F \iff u \in F^\perp$

I'm having trouble proving that in a product space $(E, \langle \cdot, \cdot\rangle)$, and subspace $F \subset E$, $\|u-v\| \geq \|v\|, \forall v \in F \iff u \in F^\perp$. In fact, the reciprocal ...
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Scalar triple product of three vectors at 60 degrees to each other

I encountered a question involving a scalar triple product of three vectors of unit length which were mutually at 60 degrees to each other. While the answer involved taking the root of the square of ...
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Finding the vertex of the parabola parameterized by $p(t)=P_0+P_1t+P_2t^2$ for vectors $P_0, P_1, P_2$

A parabola can always be described in parametric form by position vector $p(t)$, $p(t) = P_0 + P_1 t + P_2 t^2 $ where $P_0, P_1, P_2$ are vectors in $2D$ or $3D$. I would like to prove that the ...
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How does this code Find the intersection point between two lines?

I've been racking my brain for a while trying to step through this. This is Unity C# code used to find the position of intersection between two lines. The full function is here Because of my usecase I ...
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Orthogonal complement of kernel of linear functional has dimension n iff T surjective

I have successfully proven that for every linear bounded map $T:H \to \mathbb{K}^n$ it holds that $\dim(\ker(T)^\bot)\leq n$ where $H$ is a hilbert space. Now, I need to prove that equality holds iff $...
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Linear Algebra Done Right: why we define the inner product between two polynomials (of degree at most m) as $<p,q>=\int_{0}^1p(x)\bar{q}(x)dx$?

In Linear Algebra Done Right (2nd Edition) equation (6.2), we defined the inner product between two polynomials as $<p,q>=\int_{0}^1p(x)\bar{q}(x)dx$. I'm wondering if there is a particular ...
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Inner product with generating functions

Let there be two real series $F=(f_0, f_1, f_2, \dots)$ and $G=(g_0, g_1, g_2, \dots)$ which satisfy $f(x)=\sum_n f_n x^n$ and $g(x)=\sum_n g_n x^n$, where $f(x)$ and $g(x)$ are given analytic ...
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Finding a Kernel Matrix on $ \mathbb{R}^n$

Let $ X = \mathbb{R}^n$ with inner product $⟨x, y⟩ = y^TQx$ for some symmetric positive definite matrix $Q$. I have a few questions about this setup and generally about the kernels. 1- A kernel is the ...
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3 votes
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An identity of inner product

I am trying to prove the following identity $$\|\psi\|^2=\|A(A+iaI)^{-1}\psi\|^2+a^2\|(A+iaI)^{-1}\psi\|^2$$ where $A$ is self-adjoint and $a$ is a real number. My approach is to rewrite $A(A+iaI)^{-1}...
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Contraction of a lightlike vector with another vector (free cordinate)

I'm trying to prove that there exist always a vector w whose contraction with a lightlike vector u ($g(u,u)= 0$) it's always different from zero: $$g(u,v)\ne 0$$ I know how to do this with coordinates,...
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How did we get $\left<u,e_1\right>\psi(e_1)+\dots+\left<u,e_n\right>\psi(e_n)=\left<u,\overline{\psi(e_1)}e_1+\dots+\overline{\psi(e_1)}e_1\right>$?

In Linear Algebra Done Right (2nd edition), in the proof of Theorem 6.45, it is written: Let $(e_1,...,e_n)$ be an orthonormal basis of $V$. Then $\psi(u)=\psi(\left<u,e_1\right>e_1+\dots+\left&...
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Non-degenarate subspace with indefinite inner product.

I'm little overshadowed with this problem. Let $S = (v_1, · · · , v_m) ⊂ V$ be a linearly independent set which spans a non-degenerate subspace W = L(S) and put $W_k = L(v_1, . . . , v_k)$ for every $...
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Gram-Schmidt process to construct orthonormal base in a finite vector space with indefinite scalar product.

Im choking with this exercise because of the indefinite scalar product. I know the process for the definite one. The first thing I'm asked to do is to check GS is still valid for indefinite scalar ...
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Inner product space - Linear algebra 2 - Positive attribute

Lets say I have: $<(a_1,a_2),(b_1,b_2)>$ = $2a_1a_2 + 4b_1b_2$ I am going to check positive in order to refute Inner product space I get $2a_1a_2+4a_1a_2$ = $6a_1a_2$ Now, can I say that if $a_1&...
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Linear Algebra Done Right (2nd edition): what's the meaning of the notation $\overline{q}(x)$ in equation 6.2?

Linear Algebra Done Right: what's the meaning of the notation $\overline{q}(x)$ in equation 6.2: $\langle q,p\rangle=\int_{0}^1p(x)\overline{q}(x)\,dx$, where $p(x)$ and $q(x)$ are both polynomials ...
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How to prove that the range of this integral operator $T$ is $L^2[a,b]$?

The operator is $Tf= g(t)+ \int_{a}^{t}(K(t,s)f(s)ds.$ The proof assumes that $g(t) \in L^2[a,b]$ so I believe I need to only prove that $\int_{a}^{t}(K(t,s)f(s)ds \in L^2[a,b]$. I started off this ...
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Partial derivative of an inner product [closed]

How do I get the partial derivative of: $$\frac{\partial }{\partial w} \langle Y, J - XDiag(w)\rangle$$
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Inner product space and convergence

We introduce $f ∈ V$ (where $V = C([0,1],\mathbb R)$), and we put $a_n = <f, e_n>$ (where $e_n$ is an orthonormal sequence in $V$). Assuming $g(x) = \sum_{n=0}^∞ a_n e_n(x)$ is continuous, we ...
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trace Inner product

Attempt I have come across this sequence of questions whilst revising for my linear algebra exam,I have tried to answer these questions (with my attempt attached) but I am in no way sure that I have ...
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Understanding the definition of inner product by Friedberg et. al.

The following definition can be found in the book: Friedberg et. al., Linear Algebra, 5th Edition, page 327. Definition. Let $V$ be a vector space over $F$. An inner product on $V$ is a function that ...
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Notation for inner product between scalar-valued and vector-valued functions

Consider $\mathbf{f}:\mathcal{S}\rightarrow\mathbb{R}^{n}$ and $\left(\phi_k\right)_{k=1}^{\infty}$, where $\phi:\mathcal{S}\rightarrow\mathbb{C}$, and $\mathcal{S}\subset\mathbb{R}^{n}$. My intent is ...
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2 votes
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Angle between vectors in $\mathbb{C}$

I know that in $\mathbb{C}$ $$\cos\measuredangle(x,y)=\frac{|Re(\langle x,y\rangle)|}{\|x\|_2\|y\|_2}.$$ Now I want to show that $$|\sin\measuredangle(x,y)|=\frac{\|y-\lambda x\|_2}{\|y\|_2},$$ $\...
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3 votes
1 answer
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Eigenvalue of PSD matrices by constrained SDP program

Given Lemma 1, I want to prove the following corollary. lemma 1 (Rayleigh Quotient): Given matrix $A \succeq 0$, \begin{equation} \lambda_{\min} (A) = \min_{x \in \mathbb R^n}\frac{x^\top A x}{x^\...
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Inner derivative of exterior product of forms

Can anyone briefly explain the derivation of ‘Leibniz rule’ for forms: \begin{equation} \iota_{\mathbf{v}}\left(\omega^{k_{1}}\wedge \omega^{k_{2}}\right) = \left(\iota_{\mathbf{v}}\omega^{k_{1}}\...
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1 vote
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Reverse spectral theorem proof

Let $T$ be a linear transformation in a finite inner product space $V$ , let $l_1,...l_k$ be different scalars , and let $P_1 \not=0,....,P_k \not=0$ linear transformations in $V$ that satisfies the ...
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Mutual Information of Vectors with Large Inner Product

If we have a joint distribution of two (complex) vectors $x,y\in \mathbb{C}^d$ of norm $1$ such that their inner product $\langle x|y\rangle$ is $1-\epsilon$, can we lower bound the mutual information ...
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Looking for an example for the projection's theorem on an inner product space?

I'm looking for an example of a non-empty, non-convex and complete subset $C$ of an incomplete inner product space $E$ such that if we apply the projection's theorem on $C$ it gives several (maybe ...
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3 answers
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Calculating sum of matrix elements squared $\sum_{j} \sum_{k} X_{j, k}^2$ using dot product

Given a 1-D vector $v$, if asked to calculate $\sum_{i} v_i^2$, one can use a dot product trick: $\sum_{i} v_i^2 = v^T v$. I have a 2-D matrix $X$, and similarly want to calculate $\sum_{j} \sum_{k} ...
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How to prove the uniqueness criteria of the inner product defined on $L^2$ space

While trying to prove the uniqueness property ($⟨x,x⟩=0$ iff $x=0$), I understand that I would arrive at a point like this: $$⟨f,f⟩= \int |f(x)|^2 dx=0.$$ Following which I have to somehow prove that $...
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