Questions tagged [inner-products]

For questions about inner products and inner product spaces, including questions about the dot product.

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Proof that $||T(v)|| = ||v|| \forall v \implies \langle U(v), U(w) \rangle = \langle v, w \rangle$ [closed]

Is there an elegant way to prove the following Statement for a linear Transformation $T: V \to V$ where $V$ is a vector space. $||T(v)|| = ||v|| \quad \forall v \in V \implies \langle T(v), T(w) \...
edlingem's user avatar
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Matrix of an Inner Product and Spectral Theorem

My linear algebra has become very rusty and now I've confused myself entirely. Let $V$ be an inner product space over an $n$-dimensional real vector space $V$. Moreover, let the set of vectors $$\...
Algebro1000's user avatar
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Tangent vectors to regular curve at constant angle to a fixed vector

Problem: Show that the tangent vectors to the regular curve $x(t)=(3t,3t^2,2t^3)$ make a constant angle ($\theta$) with the vector $a:=(1,0,1)$. We have $b:=x'(t)=(3,6t,6t^2)$. I worked out that $b\...
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Orthogonal orthornomal bases imply pair-orthogonal vectors

While self-studying linear algebra i started thinking about following problem: Let's say that $A, B \in \mathbb{C}_{n\times n}$ are orthogonal in a Frobenius sense orthonormal bases of complex vector ...
Freechoice guy's user avatar
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2 answers
158 views
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Proving $\|v\|^2 \ge \sum _{i=1}^n \langle v,e_i\rangle^2$

Prove that: $\|v\|^2 \ge \sum _{i=1}^n \langle v,e_i\rangle^2$ for any $v \in V$, where $V$ is an inner product space and $S = \{e_1, e_2, \ldots , e_n\}$ is an orthonormal subset of $V$. I know ...
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Space of square-integrable functions and its scalar product [closed]

For square integrable functions we define a scalar product: $$\int{d^3x \psi^* (\vec{x}) \phi (\vec{x})}$$ for any $\psi, \phi \in L^2(V) $. How can I show that the fundamental property $(\psi,\phi)=(\...
Dayane 's user avatar
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Inner product in RKHS

I am reading a paper and am confused by an expression about the inner product. It says that "Given a scalar-valued RKHS $\mathcal{H}$ with a positive definite kernel $k(x,x')$, $\cdots$ and $<\...
user1168149's user avatar
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If a symmetric matrix between two vector generates a scalar, the transpose will generate the same scalar? [closed]

I have this expression, x'H0x0, where: x is a vector of variables, H0 is a symmetric matrix (Hessian) x0 is another vector. All non transposed vectors are column, all transposed vectors are rows I was ...
Hodmezor's user avatar
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Does $X \times \mathbb{R} \simeq X$ hold for infinite dimension inner product space $X$?

The $X$ is infinite dimension real inner product space, not restricting it as a Hilbert space. This question has troubled my friend and me a lot of days. It is obviously true when $X$ is infinite ...
CTuser_103's user avatar
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Is the Ball-Multiplier for the Fourier Transform auto-adjoint?

Let $f\in L^2(\mathbb{R})$, we define the ball multiplier as the operator $$Sf(x) := \int_{|\xi|<1} \hat{f}(\xi)e^{2\pi i x\cdot\xi}d\xi $$ where $$\hat{f}(\xi) := \int_{\mathbb{R}}f(y)e^{-2\pi i y\...
Mr_ion77's user avatar
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Prove $((x_n, y_n))$ converges to $(x, y)$ $\in X × Y$ if and only if $(x_n)$ converges to x in X and $(y_n)$ converges to y in Y.

$(X, ⟨·, ·⟩′)$ and $(Y, ⟨·, ·⟩′′)$ are inner product spaces over the same field, $F$. $X × Y$ is a vector field over $F$ with addition and scalar multiplication defined by $(x,\,y) + (x_1,\,y_1) = (x +...
number8's user avatar
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The positive/negative sign in inner product involve diagonal matrix and basis transformation matrix.

The expression of one inner product is $<\pi PDEP^{-1}, \pi PD^2EP^{-1}>$. Here, $\pi$ is a row vector that has strictly non-negative terms, D is a diagonal matrix with non-positive diagonal ...
Puning Wang's user avatar
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Why is vector c defined as '-s+r' [closed]

I have started following a course on coursera to learn linear algebra , In the video he defined vector c as '-s+r' or 'r-s',Why is it so. Is there any intuitive reason behind it as I can't see the ...
Delegate Of Chile's user avatar
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Effect of Basis Change on Absolute Vector Magnitudes

Does Basis Change Affect the Absolute Magnitude of Vectors? How does a change in basis impact the absolute length of vectors? I'm trying to understand the effect of a basis change on the absolute ...
fatFeather's user avatar
6 votes
2 answers
277 views

Is a scalar presented as a matrix or not here?

In the linear algebra course I am taking, the inner product of 2 vectors $\langle u, v \rangle$ is defined as being a scalar; however, it is also viewed as being a product of 2 matrices as $u^Tv$, as ...
Princess Mia's user avatar
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How should we characterize the relationship between two matrix representations of a linear operator with respect to two different orthonormal bases?

Nielsen / Chuang remark on page 71 of "Quantum Computation and Quantum Information" that, if $| v_i \rangle$ and $|w_i \rangle$ are orthonormal bases, then the operator $U$ defined by $\sum_{...
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Continuity on countably-normed Hilbert spaces

i was studying some Quantum Mechanics from this doctorate's work http://galaxy.cs.lamar.edu/~rafaelm/webdis.pdf and ata certain point, in Proposition 2 pag. 166 he means to prove the continuity of an ...
Marco Lugarà's user avatar
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1 answer
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How can an identical vector have a lower dot product value than two different vectors?

Given two vectors: $\mathbf{v}_1 = (2, 2)$ and $\mathbf{v}_2 = (3, 3)$: Since the dot product is one method for measuring the similarity between vectors, and given that $\mathbf{v}_1 \cdot \mathbf{v}...
Edmar Miyake's user avatar
4 votes
2 answers
70 views

If $U $ is a unitary linear operator, how can I show that any matrix representation of $U$ must be a unitary matrix?

Nielsen / Chuang "Quantum Computation and Quantum Information" states on p. 70: "A matrix $ U$ is said to be unitary if $U^\dagger U = I$. Similarly, an operator $U$ is unitary if $U^\...
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To find number of pairs of non-orthogonal vectors in a given set

I am quite stuck on this question. Any ideas/hints on how to approach will be helpful. Q. Let $E$ be an $n$-dimensional inner-product space over $\mathbb{R}$. Let $〈\cdot,\cdot〉$ denote the inner ...
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Can be use $u$-substitution for calculating the adjoint of an operator in Schwartz space?

I only have seen that for calculating the adjoint of an operator in $\mathcal S$, it used integration by parts, but I was thinking that if one can use substitution to find the adjoint. For exmple, for ...
Daniel Muñoz's user avatar
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Is this summation over Fourier coefficients a meaningful way to compute dot product despite its weaknesses? Or what is the better way to define it?

So let's say we have two vectors $a \in \mathbb R^n$ and $b \in \mathbb R^m$. Let $L=\text{lcm}(n,m)$. Consider the natural inclusions $a,b \mapsto \mathbb R^L$, which simply continually concatenate $...
Snared's user avatar
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Solving progressive tax calculation for pre-tax income

Progressive Tax Rate Explanation Progressive taxation works by taxing income within a certain bracket at different rates. For example: Bracket # % tax rate within bracket Min amount (exclusive) Max ...
Josh Sherick's user avatar
2 votes
1 answer
26 views

Strict chromatic vector coloring of $K_n$

In a finite simple graph $X$, for any $t\in\mathbb{R}$, a vector $t$-coloring of $G$ is a mapping $\phi_t: V(X)\longrightarrow S^m$ for some $m\in\mathbb{N}$ (where $S^m$ is the $m$-sphere in $\mathbb{...
Sanae's user avatar
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Are $\operatorname{Tr}(AB)$ and $\operatorname{Tr}(AB^T)$ bilinear forms? inner products? [duplicate]

Suppose we have $A,B \in \operatorname{Mat}(n, \mathbb{R})$, and let $\alpha(A,B) = \text{Tr}(AB)$ and $\beta(A,B) = \text{Tr}(AB^T)$. Then is it possible to show that both $\alpha$ and $\beta$ cannot ...
John Doe's user avatar
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An example of non-strictly positive-definite inner product

Give an example of non-strictly positive-definite inner product on an arbitrary vector space. By non-strictly positive-definite inner product, I mean that $||X||=0$ does not necessarily imply $X=0$. ...
schneiderlog's user avatar
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3 answers
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Finding vector equation of a line

Show that the equation of a straight line passing through the point with position vector $\vec{b}$ and perpendicular to the line $\vec{r}=\vec{a}+\mu \vec{c}$ is of the form $\vec{r}=\vec{b}+\beta \...
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Intuition of the relation between the fourier transfer and inner product of functions

Veritasium Fourier transfer video The Remarkable Story Behind The Most Important Algorithm Of All Time explains the adding up (integral) the product of the wave in interest (blue) and a known sine (or ...
mon's user avatar
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2 answers
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Find the signature of a reflection.

I am doing some self-study and I have this task: Show that the signature of the following mapping, the reflection about the hyperplane $a^{\perp}$, given by $S_a(v) = v - 2\frac{<v,a>}{<a,a&...
Newbie1000's user avatar
2 votes
0 answers
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Hilbert Spaces from Dagger Categories

Dagger compact closed categories are commonly said to be an abstraction of Hilbert spaces and is suppose to capture concepts such as unitary maps, scalars, basis, inner products. See for example the ...
Nanoputian's user avatar
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Transforming a sesquilinear form into an inner product?

I know that a vector space equipped with a seminorm can be transformed into a normed space by taking the equivalence classes of the equivalence relation $f \sim g$ iff $\|f-g\| = 0$. Can a similar ...
InMathweTrust's user avatar
1 vote
1 answer
27 views

Inner Product of Error Vector and Residual Vector

I have that $A$ is a positive definite matrix and $b$ is some fixed vector. Define $r=b-Ax$ to be the residual vector and $e=A^{-1}b-x$ to be the error vector for all vectors $x$. I need to show that $...
slaint1027's user avatar
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1 answer
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Proving that this is a convex set

Define the set $$A=\{\langle x,y\rangle : x\in [-1,1]^d, \: y\in \mathbb{R}^d, \:\|y\|_1\leq c\}$$ for some $c\in \mathbb{R}$. I want to show that is is convex. Take $\langle x,y\rangle, \langle x',y'\...
kubo's user avatar
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Why does conjugating one of the complex vectors in the inner product, still give the correct dot product?

In the complex space, I understand that we need to conjugate one of the vectors in the dot product to avoid getting that <v,v> ≤ 0. What I am unable to understand, is how taking the dot product ...
Kristoffer Eide's user avatar
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1 answer
49 views

Unknown Dot notation for vector functions regarding waves across a surface. [closed]

I came across this notation where the equations were related to waves across a liquid surface as a consequence of moving bodies within that liquid. Of course I'm familiar with dot products, but this ...
Andrew's user avatar
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2 votes
1 answer
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Projection onto a subspace of $\mathbb{R}^n$

Suppose $\mathcal V\subset \mathbb{R}^n$ is a subspace and $\vec{x}\in\mathbb{R}^n$. I want to show there is a unique way to write $\vec{x}$ as a sum of two vectors $\vec{x}=\vec{x}^{\parallel}+\vec{x}...
Bifton Mifts's user avatar
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Is $X^T Y X$ Appropriately Called a Gram Matrix if $Y$ is Positive Definite?

The Gram matrix (or Gramian) of a set of vectors in an inner product space is defined as the Hermitian matrix of inner products, with entries given by $G_{ij} = \langle \mathbf{v}_i, \mathbf{v}_j \...
Burak's user avatar
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Examples of orthogonal function bases

These days we've been solving the heat equation in class for the $1$D case of a bar of length $\ell$ with two thermal reservoirs at its ends which have the same temperature, $0^\circ$C. Yesterday we ...
Conreu's user avatar
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1 answer
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Is $\langle x,y \rangle=\overline{\langle y,x \rangle}$?

can anyone help me? Be $(X,\|.\|)$ a normed space that satisfies the parallelogram law. Let's define $\Phi : X\times X \rightarrow \mathbb{C}$ by: $\Phi (x,y)=\dfrac{1}{4}[ \|x+y\|^2-\|x-y\|^2+i\|x+...
Luis Amando Melendez Rojas's user avatar
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2 answers
51 views

Dimensionality of the set of linear maps between inner product spaces

Suppose we have two inner product spaces $V$ and $W$ with dimensions $n$ and $m$, respectively. For some set $\{ \mathbf{v}_i\}_{i=1}^k \in V$ and some nonzero $\mathbf{w} \in W$, where $1 \leq k \leq ...
Goentagen's user avatar
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1 answer
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Demonstrating equality of Extended Cauchy-Schwarz Inequality [closed]

I have a bit of a problem with the Extended Cauchy Schwarz Inequality, specifically at the line "with equality if and only if b = cB^(-1)d for some constant c." As such, I'm having problems ...
depsilon01's user avatar
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Matrix associated with linear transformations of the form $T(v):=\Sigma _{i=1}^ma_i\langle v,v_i\rangle w_i$

Let $V,W$ be two finite dimensional Hilbert spaces over $\mathbb{R}$. Suppose that $\{v_1,\cdots,v_m\}$ and $\{w_1,\cdots,w_n\}$ are orthonormal subsets of $V$ and $W$, respectively. Define $T:V\to W$...
noob's user avatar
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Product of vector transpose and matrix

Let $A$ be a $m\times n$ matrix and $\boldsymbol{v}$ be a $n$-dimensional vector and $\boldsymbol{u}$ be a $m$-dimensional vector. I have difficulty understanding $\boldsymbol{u}\cdot (A\boldsymbol{v})...
Ray Siplao's user avatar
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An equivalent condition for isometries

Despite the long physical introduction, I swear this is a mathematical question. While introducing the motion of a point seen by two observers O and O' (with respective Euclidean spaces E3 and E3'), ...
Davide Masi's user avatar
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19 views

What are All the Reflections in Minkowski Space $\mathbb{R}^{1,n}$?

All the literature on reflections in minkowski space, that I have found, have defined ways to reflect about an arbitrary planes or lines and they always add the disclaimer eventually that the plane or ...
intravertig0's user avatar
7 votes
2 answers
122 views

On the definition of quadratic forms

Let $\mathbb{F}$ be an arbitrary field of characteristic $\mathrm{char}(\mathbb{F})\neq 2$ and $V$ a $\mathbb{F}$-vector space. The definition of a quadratic form I am used to is a map $\varphi\colon ...
G. Blaickner's user avatar
5 votes
2 answers
348 views

The projective tensor norm on tensor product of Banach spaces implies the inner product on tensor product of Hilbert spaces?

As presented in the answer of this post, the projective tensor norm on the algebraic tensor product of two Banach spaces $X$ and $Y$ is given by \[ \Vert \omega\Vert_{\pi} = \inf\left\{\sum \lVert x_{...
Keith's user avatar
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3 votes
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Why is the order reversed in the conjugate symmetry axiom for the inner-product?

The conjugate symmetry axiom for the inner product is as follows: $$\overline{\langle x,y\rangle } = \langle y,x\rangle$$ It is my understanding that the conjugate is there so that the norm of a ...
cookiecainsy's user avatar
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Inner product symmetry axiom confusion with QR compositions

So I'm currently doing an advanced linear algebra module and I'm confused on two statements from two different lectures. I'll attach a section of two lectures and highlight the statements that ...
Stefan W. Vorster's user avatar
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1 answer
28 views

Inequality regarding inner product and functions with zero integral

Suppose $X$ is a finite set and $f:X\rightarrow \mathbb{R}$ satisfies $\sum_{x\in X}f(x)=0$. Let $p\in\Delta(X)$ be a probability measure on $X$. Does the following statement hold? $$ \sum_{x\in X} f(...
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