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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.

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Kernel function with a feature space equipped with an inner product that is not the dot product

Premise: A function $K: \mathbb R^d \times \mathbb R^d \to \mathbb R$ is called a kernel function on $\mathbb{R}^d$ if there exists a Hilbert space $\mathcal{H}$ and a map $\phi: \mathbb R^d \...
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Proof no nonnegative function satisfies three norm-like integrals

$V$ is the real inner product space $C_\mathbb{R}[0,1]$ and $a \in [0,1]$. Prove there is no nonnegative function $f \in V$ such that $$ \int_0^1 f(x) \, dx = 1, \int_0^1 xf(x) \, dx = a, \mathrm{and} ...
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Normalizing dual eigenvectors? Why only for trivial defects?

Two little questions to this passage: (1) How can we normalize to get $\langle u,u^*\rangle =1$? (2) Why is this possible if $\lambda$ has trivial defect only (i.e. for trivial Jordan blocks)? I did ...
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Are there degrees or categories of orthogonality?

The vectors $\begin{bmatrix}a \\ 0 \\ 0\end{bmatrix}$, $\begin{bmatrix}0 \\ b \\ 0\end{bmatrix}$, and $\begin{bmatrix}0 \\ 0 \\ c\end{bmatrix}$ are orthogonal under the dot product. The vectors $1$, $...
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Real Inner Product Space, Hermitian Operator $T = S^{n}$ for n odd

Let V be a finite dimensional inner product space over $\mathbb{R}$, and $T: V\rightarrow V$ hermitian. Suppose n is an odd positive integer. Want to show: $\exists S:V\rightarrow V $ such that $T = ...
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With these definitions how do I prove that this inner product is positive-definite?

Could someone please help me untangle the notation with the following? Let $V$ be a real vector space with inner product $\langle \cdot \,,\cdot \rangle : V \times V \rightarrow \mathbb{R}$. Let $W = ...
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$PQ = 0 \Leftrightarrow P(H) \bot Q(H)$ if $H$ is Hilbert and $P,Q$ orthogonal projections.

Exercise : Let $H$ be a Hilbert space and $P,Q \in \mathcal{L}(H)$ orthogonal projections. Show that : $$PQ = 0 \Leftrightarrow P(H) \bot Q(H)$$ Attempt-Thoughts : $(\Rightarrow)$ Let $PQ = 0$. ...
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Prove that $W$ has dimension $n$ over $\mathbb{C}$, assuming that $V$ has dimension $n$ over $\mathbb{R}$.

Let $V$ be a real vector space with inner product $\langle \cdot \,,\cdot \rangle : V \times V \rightarrow \mathbb{R}$. Let $W = V \times V$ with vector addition defined by $(v_1,w_1) + (v_2,w_2 ) = (...
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There is a General form for Inner Product?

I am studying Linear Algebra and my professor proposed an exercise: Let $\mathbb{R^n}$ a vector space, $v=(v_1,...,v_n)$ and $u=(u_1,...,u_n)$, then there is a general form for Inner Product between $...
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Two inner products different by a scalar

Let $\langle \cdot, \cdot \rangle_{1}$ and $\langle \cdot,\cdot \rangle_{2}$ be inner product on a finite-dimensional vector space with the property that \begin{align*} \langle u,v\rangle_{1}=0 \...
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Showing scalar product properties on certain matrix multiplication

from S.L Linear Algebra: Let $M$ be a square $n \times n$ matrix which is equal to its transpose. If $X$, $Y$ are column $n$-vectors, then: $$X^TMY $$ is a $1 \times 1$ matrix, which we ...
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Triangle Inequality for Angles in Projective Space

I want to show that the angle between two lines through the origin in a (complex or real) inner product vector space $(V,\langle \cdot,\cdot\rangle)$ is a distance function which turns $\mathbb{P}V$, ...
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Assume $\langle v,s \rangle + \langle s,v \rangle \leq \langle s,s \rangle$

The question is Let $V$ be a complex inner product space, and let $S$ be a subspace of $V$. Suppose that $v\in V$ is a vector for which $\langle s,v\rangle + \langle v,s\rangle \leq \langle s,s\...
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Show that graph of operator with adjoint operator is closed

Let $X,Y$ be inner-product spaces. Let $T\in L\left(X,Y\right)$ be a linear operator with adjoint operator $S\in L\left(Y,X\right)$ such that $$\langle Tx,y\rangle_Y=\langle x,Sy\rangle_X\quad\forall (...
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Maximize inner product subject to constraint

Let $a\in\mathbb{R}^{d}$ and $K$ be an arbitrary subset of $\mathbb{R}^{d}$. My question is related to the following optimization problem: \begin{equation} \max_{x\in\mathbb{R}^{d}}~a^{\top}x\quad \...
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Computing the inner product $(\chi_{R},\chi_{R})$.

Let $\chi_{R}$ denote the character of the right regular representation $R$. Compute the inner product $(\chi_{R},\chi_{R})$ directly and also by decomposing $R$ into a sum of irreducible ...
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Examples of non-unitary isometries on finite dimensional Hilbert spaces?

I was reading the question A Finite Dimensional non-Unitary Isometry?, which gives an example of a non unitary isometry which is a map $T: R \rightarrow R^2 $. This question is based on a previous ...
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Fourier Series Exponential Representation

As part of my education, I took upon myself to understand where the Fourier series functions come from, I did some digging, and found out that the vector space accommodating the function is a Hilbert ...
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How is Schwartz space different from Hilbert space?

I know that Schwartz space can be considered a dense subset of the Hilbert space isomorphic to $\ell^2$. What I wish to understand is, how really different Schwartz space is from the Hilbert space. ...
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Other inner products for $\mathbb{R}^n$

For $\mathbb{R}^n$, the standard inner product is the dot product. It is defined as $ \langle v,\,w\rangle = \sum_i v_i \cdot w_i $. I am aware that any scaled version, namely $ \langle v,\,w\rangle ...
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If $A$ is symmetric, why is $ x^TA^Ty=(x^TA^Ty)^T $ obvious?

The question has emerged when I read the proof of the following Theorem: Let $V$ be a $\mathbb{R}$-vector-space, with $\dim(V)=n< \infty, C=(c_{1},...,c_{n}) $ a$ $ Basis of $V$, and $B$ a ...
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Show that A is skew-symmetric if and only if $x^tAx = 0$

I've tried by starting with setting $x^tAx = 0 = x^t(-A^t)x$ and checking it termwise, but I don't think this will show me anything. Could you explain how to approach this problem please?
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Definition of outer product

I am trying to understand the concept of outer product in quantum mechanics. I read "Quantum Computing explained" of David MacMahon. I can understand the transition in (3.12): $$(|\psi\rangle \...
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Does Cauchy-Schwarz hold for: $ \langle\textbf{u},\textbf{v}\rangle \;\leq ||\textbf{u}|| \cdot ||\textbf{v}|| $

I am wondering whether the Cauchy-Schwarz inequality does hold when absolute value is not considered for the LHS. Let me explain: In standard Cauchy-Schwarz we have: $| \langle \textbf{u},\textbf{v}...
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Matrix-based proof of transformation rule for a wedge product of covectors by wedge product of covectors

I am supposed to show $$\beta^1 ∧ ··· ∧ \beta^k = (\det A) \gamma^1 ∧ ··· ∧ \gamma^k$$ for covectors $\beta^1, ···, \beta^k, \gamma^1, ···, \gamma^k$ of a vector space finite dimensional ...
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Derivations of bases of $A_k(V)$ and $L_k(V)$: what's the difference?

My book is An Introduction to Manifolds by Loring W. Tu. Here is the derivation of a basis for $A_k(V)$: Proposition 3.27, "Wedge product of 1-covectors" Lemma 3.28, Proposition 3.29 Proposition 3....
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Describe $f$ in terms of the tensor products of $\alpha^i$ and $\alpha^j$. Is this inner product? What are its coefficients?

My book is An Introduction to Manifolds by Loring W. Tu. In an exercise, we are asked to describe a bilinear function in terms of tensor products. The answer is given at the back is $f= \sum g_{ij} \...
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Show {$x^k$}$_{k\geq 0}$ is complete in $L^2[a,b]$

I'm thinking of using the Weierstrass Approximation Theorem where the span of monomials is dense in the continuous functions.
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$V$ is a inner product space, prove $\langle av, v\rangle \langle v, av\rangle \le \langle av, av\rangle $

Here is the problem $V$ is an inner product space, and a is a linear transformation from $V$ to $V$. Prove that for any unit vector $v$ belongs to $V$, we have $\langle av, v\rangle \langle v, av\...
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Is inner product preserved on change of basis?

Suppose we have a matrix $A$ over which an inner product is defined. Let us denote this inner product by $\langle, \rangle_A$. Now let us suppose that this matrix $A$ is symmetric. Then there exists ...
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Cauchy-Schwarz inequality for $L^2$-norm on periodic functions space

I have proven something that is definitely not true (Lemma 2), which is why I am intersted where I err. Definition Let $C(\mathbb{R}/\mathbb{Z},\mathbb{C})$ be the set of all continuous $\mathbb{Z}$-...
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Inner product inequality involving linear operator

$V$ is an inner product space and $\tau \in \mathcal {L}(V)$. I want to show that for any unit vector $v\in V$, $\langle \tau v, v\rangle\langle v,\tau v\rangle\leq\langle \tau v,\tau v\rangle$. I ...
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simplifying complex inner product by factoring out complex constants

From "Linear Algebra Demystified", David McMahon, 2006, problem 2, page 132 and page 235: $$<\underline{u},\underline{v}> = 2i$$ $$<\underline{u},\underline{w}> = 1 + 9i$$ $$ y = (2\ <...
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How to prove the inner product of complex vectors is conjugate-symmetric? [closed]

For complex vector-space of 2 dimensions, prove that the inner product is conjugate-symmetric, ie: $$ < \underline{x},\underline{z}> = <\underline{z},\underline{x}>^{*} $$ where: $$ \...
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Inner products harmonic oscillator

How do I compute these inner products; $(\Delta x)^2 = \frac{\int_R \!x^2\psi_m\psi_mdx}{\int_R \!\psi_m\psi_mdx}$ $(\Delta p)^2 = \frac{\int_R \!\psi_m\frac{d^2}{dx^2}\psi_mdx}{\int_R \!\psi_m\...
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Prove $l^2$ norm obeys the triangle inequality

I'm trying to work through Exercise 3 from this blog post, which is essentially a proof of the validity of the $l^2$ norm: Exercise 3: Let $(\mathcal{V},\left<\cdot,\cdot\right>)$ be an inner ...
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What is the minimal structure required to talk about “directionality” and “angles”?

Intuitively speaking, vector spaces are inherently endowed with a concept of “directionality”, since a vector is intuitively an arrow in some direction. But if I’m not mistaken, we need to endow the ...
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1answer
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Can we define an inner product in terms of the norm induced by it?

I know that not all norms are induced by any inner product. But if we have an inner product, $\langle\cdot,\cdot\rangle$ we can define a norm $||v||=\sqrt{\langle v,v\rangle}$. My question is, can ...
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Defining a generalised version of inner product over $*$-fields

I see definitions (in Wikipedia, for example) about inner product spaces over arbitrary fields. But I don't understand how positivity makes any sense for fields which are not ordered? Am I missing ...
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Inner product with values in a finite field over space of finite functions

Suppose we have $\mathbb{Z}_n$ the group of residues modulo $n$ and $\mathbb{F}_q$ a Galois finite field with $q$ elements where $q=p^m$ with $p$ prime and $m\in\mathbb{N}$ and suppose $n\vert q-1$. ...
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A construction of the Hodge Dual operator

This question about showing that an alternative construction of the Hodge dual operator satisfies to the universal property through which the Hodge dual is usually defined. Let me give the ...
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Integral of inner product of $\langle x, A(t) \dot{x}\rangle$

Let $A(t)$ be uniformly positive definite for any $t$, and $x(t)$ is a vector, and $x(0)$ is a finite. since $\int_0^t x(\tau)\dot{x}(\tau)\,\mathrm d\tau = \frac{1}{2}[x(t)^2-x(0)^2]$ is finite, I ...
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About the Definition of Preserving Inner Products $ (T \alpha|T \beta)=(\alpha|\beta) $

In the above definition, when it says $ (T \alpha|T \beta)=(\alpha|\beta) $, the inner product $(|)$ acting on $W$ is the same as acting on $V$? Or not necessarily? The book uses the same notation, so ...
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Cauchy Schwarz inequality with 1 norm

Here is the argument I am making By Holder's inequality, we have for $\frac{1}{p} +\frac{1}{p^*} = 1$ $$\langle A, B\rangle \leq ||A||_p||B||_{p^*}$$ The Schatten p-norms also obey $||A||_p \geq ||...
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Inequality involving inner product and norm

If $\|\cdot \|$ is the norm induced by the inner product $\langle,\rangle$, how to prove the following interesting inequality? $$\langle x,y\rangle(\|x\|+\|y\|) \leq\|x+y\|\,\|x\|\,\|y\|$$ This is a ...
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Definite integral as dot product

I was reading examples of dot products and I came across the following: In the vector space $P_n(\mathbb{R})$ of polynomials with degree less or equal to $n$ we consider the following dot product:$$...
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Completion of a subspace of a Hilbert space with respect to a positive semi-definite symmetric bilinear form

Let $H$ be a $\mathbb R$-Hilbert space $\mathcal E$ be a positive semi-definite symmetric bilinear form on a dense subspace $\mathcal A_0$ of $H$ and $$\mathcal E(f):=\mathcal E(f,f)\;\;\;\text{for }...
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Weighted inner product with arbitrary matrix?

An inner product can be written in Hermitian form $$ \langle x,y \rangle = y^*Mx $$ that requires $M$ to be a Hermitian positive definite matrix. I have read that using Hermitian positive definite ...
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Show that $f_A$ is an inner product

Let $A$ be a $2 \times 2$ matrix with real entries. For $X, Y$ in $R^{2 \times 1}$ let $f_A(X, Y) = Y^tAX$. Show that $f_A$ is an inner product on $R^{2 \times 1}$ if and only if $A = A^t$, $A_{11} &...
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Show {$e^{2\pi i bnx}$} is a basis for $L^2[0,b^{-1}]$

Let $b > 0$ be a fixed positive scalar. Show {$e^{2\pi i bnx}$} for $n \in \mathbb{Z}$ is a orthogonal (but not orthonormal) basis for $L^2[0,b^{-1}]$. I was able to show it is orthogonal and not ...