Questions tagged [inner-products]
For questions about inner products and inner product spaces, including questions about the dot product.
4,916
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The distance in $\mathbb{R}^n$ is not induced by an inner product.
We define the distance like $d(v,w)$ = {the number of different entries} and I'm supposed to prove that this is not induced by an inner product. So far what I´ve done is use the parallelogram law to ...
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Understanding Matrix Multiplication: Column-Row Decomposition
I recently started reading a college linear algebra class while in middle school. I am reading section 1.3, which states that given matrices $ \mathbf{A} \in \mathbb{R}^{m,k} $ and $ \mathbf{B} \in \...
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Show that $(Ker(\psi))^{\perp} \subset Im(\phi)$
I'm trying to solve the following problem:
Let $V$ and $W$ be two finite dimensional vector spaces over $\mathbb{C}$ with inner products $p$ and $q$ respectively. Let $\psi: V \rightarrow W$ be a ...
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Mock scalar product's nonlinearity as a derivation
Context. I was reading about the converse of the parallelogram law (a norm satisfying the law defines an inner product), and in an answer on MO it is stated that this is impossible without the norm's ...
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How to understand this equation by linear algebra?
Assume that
$$
\begin{pmatrix}
x' \\
y'
\end{pmatrix} =
\begin{pmatrix}
u & Dv \\
v & -u
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
$$
with $D>0$,
then we have
$$
x'^2+Dy'^2 = (...
- 141
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dot product of surface and gradient
I've been asked to answer if the following dot product make sense: $F \cdot \nabla f$, when $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ is $C^1(\mathbb{R}^3)$ and $F: \mathbb{R}^3 \rightarrow \mathbb{R}^...
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If $U$ and $V$ are unit vectors and $U\cdot V=1$, is it true that $U=V$?
EDIT: without using $U\cdot V=\lVert U\rVert \lVert V\rVert\cos{\theta}$,
If $U$ and $V$ are unit vectors and $U\cdot V=1$, does that mean $U=V$? I know $U\cdot V=U\cdot U=V\cdot V=1$. But I also know ...
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Showing that $(XY_n)$ is a Cauchy sequence when $(Y_n)$ is cauchy in space of polynomias.
Letting $\mathbb{C}_0[X]$ be the space of complex polynomials without constant term and $(\kappa_n)$ a sequence of real numbers (To be precise, the $\kappa$'s are the free cumulants of a compactly ...
- 154
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What is wrong in this Schwarz inequality？
I have a inequality in a book's problem. It is Let $ A $ be a positive definite Hermitian operator.Show that for all $ \left|u\right\rangle $ and $ \left|v\right\rangle$,$$|\left<u|A|v\right>|^2\...
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expectation of sum (or square sum) of two matrices which entries are sampled from normal distribution
Suppose we have two matrices, $A \in \mathbb{R}^{n \times k}$ and $B \in \mathbb{R}^{m \times k}$, and each entry of the matrices are sampled from $N(0, 1)$, I want to compute the expectation and ...
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What is the adjoint of the curl operator?
Let $\Omega \subseteq \mathbb{R}^3$ be a bounded, connected domain, and set $$\mathcal{V} = \{ \vec\phi = (\phi_1, \phi_2, \phi_3) \in C_c^\infty(\Omega): \nabla\cdot \vec\phi=0\}.$$ Denote $V$ to be ...
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Linear algebra object unclear in the theory behind PCA
Basic issue but one that created a moment of reflection for me. In studying PCA we use the unit vector $\vec{w}_{1}$ and reproject all observations $\vec{x}_{n}$ onto this via the formula:
$\vec{w}_{1}...
- 235
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If $T$ is self-adjoint and $\alpha \in \mathbb{R}$, show that $\alpha \langle Tv,v \rangle \geq -|\alpha| \lVert Tv \rVert \lVert v \rVert$
I'm studying Sheldon Axler's "Liner Algebra done right" book, but I'm having some trouble understanding the proof of Lemma 7.11.
Lemma (7.11): Suppose that $T \in \mathcal{L}(V)$ is self-...
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Any inner product is determined by the lengths of vectors
For any complex inner product space $V$ and for any $u,v \in V$, we have:
$$4 \langle u,v \rangle = \| u+v\| ^2 - \| u-v\|^2 +i\| u+iv\|^2 -i\| u-iv\|^2$$
to which my lecture notes conclude: "any ...
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Comparison of the expectation of minimums of random variables
I'm examining the relationship between the expected values of the minimums of three independent, non-negative random variables $X$, $Y$, and $Z$, such that $E[2X] = E[Y] + E[Z]$. Specifically, I want ...
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For a compact group $G$ and any finite-dimensional complex representation $(\pi,V)$, how to show that $V$ admits an invariant inner product?
To demonstrate that an inner product $\langle\ ,\ \rangle$ on $V$ is invariant, one need to check
$$\langle\pi(g)v,\pi(g)w\rangle=\langle v,w\rangle.$$
For an arbitrary inner product $\langle\langle\ ,...
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Proof that an operator is self-adjoint if and only if its matrix is self-adjoint
As stated in the wikipedia page on self-adjoint operators, $A$ is a self-adjoint operator on an finite-dimensional inner product space $V$ if and only if, given an orthonormal basis, the matrix of $A$ ...
- 4,708
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Bound on $|x \cdot y|$ from below
Take $x$ and $y$ to be real $n$-dimensional vectors with an angle $\theta$ between them. Also take "$\cdot$" to mean the real dot product. We can bound $|x \cdot y|$ from above pretty easily ...
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How to get a vector from two points
I am trying to find the bond angle between two vectors in a tetrahedron that is, the angle between BE and DE
however seems like the vectors are wrong.
Those vectors we find subtracting points
$$E = (...
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What surface is represented by $\mathbf{r}\cdot\mathbf{a} = \mathrm{const.}$?
What surface is represented by $\mathbf{r}\cdot\mathbf{a} = \mathrm{const.}$ that is described if $\mathbf{a}$ is a vector of constant magnitude and direction from the origin and $\mathbf{r}$ is the ...
- 93
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The correct way to map a column vector into a row vector. [closed]
I'm a physics student studying linear algebra and the matrix representation of usual and dual vectors. I have several questions related to this topic, and I'm not sure if they make sense.
Firstly, the ...
- 13
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Doubt in proof of Jordan - von Neumann theorem
In Paliogiannis and Martin A. Moskowitz's Functions Of Several Real Variables, the Jordan - von Neumann theorem is given as follows:
The inner product is defined as: $$\langle x,y \rangle = \frac{||x+...
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What do we measure using norm?
Over the course of my studies I have acquired quite rigid notion of distance (probably due to my interaction with Physics). So when studying vector spaces, more specifically $\mathbb {R}^n$, I can't ...
- 165
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Show $A \subset B \Leftrightarrow B^{\perp} \subset A^{\perp}$ with $A,B$ subsets of a Hilbert space.
Show $A \subset B \Leftrightarrow B^{\perp} \subset A^{\perp}$ with $A,B$ subsets of a Hilbert space.
The forward direction is easy: Assume $A \subset B$. For any $a \in A$ and $x \in B^\perp$ since $...
- 2,609
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Find the matrix maximizing a summation of bilinear forms
Given $d, n \in \mathbb{N}$ and $x_k, y_k \in \mathbb{R}^d$, for $1 \leq k \leq n$,
we want to find
$\arg \max_{A: ||A||_2 = 1} \sum_k \langle x_k, A y_k \rangle$,
where $||\cdot||_2$ denotes spectral ...
- 1,353
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Show that an inner product space is always strictly convex.
Just looking for feedback on the proof for this problem, any feedback is appreciated.
The full problem is as follows:
A normed vector space V is strictly convex if $\left\lvert\left\lvert u \right\...
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Dependency of the Hessian on the inner product
Question: the gradient of a function $f:E\rightarrow\mathbb{R}$, with $E$ being a vector space equipped with an inner product $\langle\cdot,\cdot\rangle$, depends on the choice of the inner product ...
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Exercise $3$, Section $6.A$ - Linear Algebra Done Right
Exercise: Suppose $F = R$ and $V \ne \{0\}$. Replace the positivity condition (which states that $\langle v, v\rangle \ge 0$ for all $v \in V$ ) in the definition of an inner product $(6.3)$ with the ...
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Does Hilbert $C^*$-modules have orthonormal basis?
Let $E$ be a (right) Hilbert $A$-module where $A$ is unital $C^*$-algebra. I call a subset $S$ of $E$ to orthonormal if $\langle x,x\rangle=1$ ($1$ denotes the unit in $A$) for all $x\in S$ and $\...
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Minimum bound of integral of $f^2$ subject to inner product constraints
I'm copying this problem from aops (the work I have done so far is also there), I wanted to get some more exposure to it from a different audience.
Given $f\in L^2[0,1]$ satisfying $\int_0^1 f(x)\, dx=...
- 5,092
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Showing $\prod_{i=0}^{n-1} (3^2+2i)=\frac{2^{n+5}\Gamma(n+\frac{11}{2})}{105\sqrt{\pi}}$
Deduce the following equality:
$$\prod_{i=0}^{n-1} (3^2+2i) = \frac{2^{n+5} \Gamma(n+\frac{11}{2})}{105 \sqrt{\pi}}$$
So I have the product by expansion:
$$\prod_{i=0}^{n-1} (3^2+2i)=3^2(3^2+2\cdot 1)(...
- 243
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Finding parallel vectors knowing orthogonal projection [closed]
Two vectors $u$ and $v$ are given as $u = i-j+k$, $v = 2i-j+3k$;
I found the angle between the two vectors using $$\cos\theta= {(vu)\over(|v||u|)}$$ therefore simple dot-product, and the result is $$\...
- 21
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A inequality satisfied over an ellipsoid
Let $A$ be a positive definite matrix and $v \in \mathbb{R}^n$ be a given vector. Suppose there exists a $y \in \mathbb{R}^n$ such that $0 < y^{\top}Ay \leq 1 $ and the following holds:
$$
\langle ...
- 113
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Conjugate symmetry in inner products and the intuition behind projections
Before I begin, I understand why inner products need to maintain conjugate symmetry as seen in this post. This is not what my question is directly about. My motivation is to derive complex Fourier ...
- 109
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Must the domain of a Hilbert space operator be limited to square-summable elements? If so, what type of space does not restrict, thus?
Background: The operation of bandlimiting is a projection operator, $P$, into a separable Hilbert space. For example, with inner product, $\int_{-1/2}^{1/2}g(f)\overline{h(f)}df$, there is an ...
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Eigenvectors of any arbitrary matrix is same as its adjoint.
Recently I came across Normal matrices and their properties, one of which states that their eigenvectors are the same as their adjoint and are orthogonal. I've gone across some proofs and I understand ...
- 109
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Expected value of the inner product equals l2-norm?
Referring to Butucea (14), I have given $(Y_i)_{i\in \mathbb N}$ iid. with Lebesgue-density $p$, a kernel $K$ and the scalar product $\langle f,g\rangle=\int_{-\infty}^\infty f(x)\overline{g}(x)\...
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Derivation of Complex Fourier Series coefficients through inner products (and swapping arugments)
I am trying to derive the complex Fourier series coefficients given by:$$f(x)=\sum_{n=-\infty}^{\infty}{c_n}e^{inx}$$ with coefficients:$$c_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}{f(x)}e^{-inx}dx$$
I am ...
- 109
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Inner product invariant under group action - meaning?
My professor wrote a proposition where he defined an inner product on a $G$-module $V$. One of the premises was that the inner product was 'invariant under the action of the group', however he did not ...
- 833
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Solving an equation in a Hilbert space
Suppose $\mathbb{H}$ is a Hilbert space over the reals. Fix two vectors $\alpha$ and $\beta$ in $\mathbb{H}$. I came across a problem that reduces to the following:
What are the possible units vector $...
- 8,593
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Is there a relationship between $\|X\|_2 ^2$ and $(\sum_i x[i])^2$?
Is there a known relationship between $\|X\|_2 ^2$ and $(\sum_i x[i])^2$, where $X$ is a vector in the real space, $\|\cdot\|_2$ is the 2-norm, and $\sum_i x[i]$ is the sum over the elements of the ...
- 97
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An algebraic abstraction of dot and cross product
In a recent physics class, we proved the expansion formula
$$ \vec{\nabla}\times(\vec{A}\times \vec{B}) = \vec{A}(\vec{\nabla}\cdot\vec{B}) + (\vec{B}\cdot\vec{\nabla})\vec{A} -(\vec{B}(\vec{\nabla}\...
- 1,226
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Does this modification of Gram matrix have a particular name?
If we have $m$ $n$-dimensional vectors $\mathbf{u}_1,...\mathbf{u}_m$, the matrix of their inner products is a Gram matrix. For real vectors $\mathbf{v}$ and $\mathbf{w}$, the inner product is the sum ...
- 414
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An inequality in a Lorentzian vector space
Let $(V,\langle \cdot,\cdot\rangle)$ be a Lorentzian $n+1-$dimensional vector space. I want to prove that, given a nonzero linear functional $f\in V'$, we have that for any two future timelike vectors ...
- 318
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Theorems of elementary geometry that can be generalized to inner product space over $\mathbb{R}$ [closed]
In inner product spaces on $\mathbb{R}$, length and angles can be defined. Hence, something similar to elementary geometry can be done there.
For example, the parallelogram law can be generalized to ...
- 1
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The Inner Product of a Hadamard Product
So let's say I have the inner product:
$$\vec{y}_1^H \vec{y}_2 = (\vec{x}\circ\vec{h}_1)^{H} (\vec{x}\circ\vec{h}_2) = \sum_{i} (x_i^{\ast} h_{1,i}^*) (x_i h_{2,i}) = \sum_{i} |x_i|^2 h_{1,i}^* h_{...
- 151
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Matrices of self-adjoint operators
Fix a finite-dimensional real inner product space. Does a self-adjoint operator have a symmetric matrix with respect to all bases, or just with respect to orthonormal bases? Also, does this change at ...
- 1,169
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Are the limits of Euclidean spaces Hilbert spaces?
Consider each Euclidean space $\mathbb{R}^n$ as a topological vector space. I wondered what I get by taking $n \to \infty$ in the category of topological vector spaces over $\mathbb{R}$.
There are ...
- 1,699
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Proof of Cauchy shwarz inequality with multiple parts
The aim of this question is to prove the cauchy schwarz inequality.
Use the fact that $\|\mathbf{x}\|^2 = \mathbf{x} \cdot\mathbf{x} \geq 0$ to show the following:
a) $\|\|\mathbf{v}\|^2 \mathbf{u} - ...
- 2,699
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$T$ and $U$ self-adjoint on $V$, $T$ is positive definite. Prove $TU$ & $UT$ are diagonalizable linear operators that have only real eigenvalues.
This is a problem from Freidberg linear algebra (4th edition) chapter 6.4.21
The whole problem and the hints are
Let $V$ be a finite-dimensional inner product space, and let $T$ and $U$ be
self-...
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