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Questions tagged [orthonormal]

For questions related to orthonormality. A set of vectors in an inner product space is called orthonormal if each vector is a unit vector, and vectors are pairwise orthogonal.

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Equivalence of statements about bounded linear maps on a Hilbert space

Assume $H$ is a Hilbert space and $V \in L_b(H)$. I want to show, that the following propositions are equivalent: V is an isometry. For every orthonormal system $\{u_{\alpha}: \alpha \in A \}$ the ...
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Can the eigenvectors of a linear operator in an infinite-dimensional space span the space and be linearly dependent at the same time?

Consider a vector space $V$ over the complex field which is infinite-dimensional with a Euclidean inner-product. Let $L$ be a linear operator on $V$. Say a subset of eigenvectors of $L$ forms a ...
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+50

Find which conditions must parameters $a$ and $b$ meet so there's exist an orthonormal basis

In $\mathbb{E^3}$ we have the plane $\pi:x-y+z-3=0$, the line $r:(2,0,1)+t(1,1,0),\ t\in\mathbb{R}$, and the point $P=(3,0,3)$. Which conditions must parameters $a$ and $b$ meet so there's exist an ...
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Inner product that makes vectors an orthonormal basis

Let $X= \begin{pmatrix} a \\ b \end{pmatrix} $ and $Y=\begin{pmatrix} c \\ d \end{pmatrix}$ be two vectors in the plane. Do we have the existence of an inner product that makes ...
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Write $(-1,1,2,2)^T$ in terms of the basis $U$

I have found that $U=\{1/2\pmatrix{1\\1\\1\\1},1/2\pmatrix{-1\\-1\\1\\1},\frac{1}{\sqrt{2}}\pmatrix{-1\\1\\0\\0},\frac{1}{\sqrt{2}}\pmatrix{0\\0\\1\\-1}\}$ is an orthogonal basis for $\mathbb{R}^4$. ...
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35 views

Orthonormal basis of eigenvectors of a self-adjoint operator

Let $V$ be a finite-dimensional real inner product space and suppose $T$ is an endomorphism on $V$. Show that $(T+T^*)/2$ is self-adjoint and show that there is an orthonormal basis $\{\vec{v_1},...,\...
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A necessary and sufficient condition for weak convergence in a Hilbert space

The problem is: Suppose $H$ is Hilbert and $\{e_n\}_{n = 1}^\infty$ is its orthonormal basis. Prove $x_n \rightharpoonup x_0$ if and only if: $||x_n||$ is uniformly bounded (i.e. $\exists M >0$ s....
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Linear algebra orthonormal basis?

Assume that A ∈ Mn×n(R) admits an orthonormal basis of eigenvectors with eigenvalues λ1 ≤ λ2 ≤ · · · ≤ λn. Show that λ1∥v∥2 ≤ Av · v ≤ λn∥v∥2 for each v ∈ Rn So far I've got: ⟨vi,vj⟩ = 0 for i= ̸=j ...
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Orthogonal bases of the vector space $\mathbb{Z}_2^4$

Let $\mathbb{Z}_2$ be the two element field $\mathbb{Z}/2\mathbb{Z}$. The vectors $e_0 = \langle1,1,1,1\rangle$, $e_1=\langle1,1,0,0\rangle$, $e_2 = \langle1,0,0,1\rangle$, $e_3 = \langle1,0,1,0\...
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Distribution of column of random orthogonal matrix

Suppose $O \in \mathbb{R}^{n \times r}$ with $r < n$ is a random matrix whose distribution is uniform on the set of $r \times n$ matrices such that $O'O = I_r$. Is is true that the columns of $O$...
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Prove the uniqueness of orthogonal function without considering sign

Suppose that the group $p_0(x), p_1(x),... p_N(x) $ are orthonormal in the interval $[-1,1]$, which means with 2 arbitrary functions $p_i(x), > p_j(x)$, the conditions below is satisfied. $\...
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Then which of the following statements are true?[CSIR-2018-December]

Let $\{u_1,u_2,..., u_n\}$ be an orthonormal basis of $\mathbb {C^n}$ as column vectors. Let $M=(u_1,u_2,...,u_{k})$ and $N=(u_{k+1},u_{k+2},...,u_{n})$ and $P$ be a $k \times k$ diagonal matrix with ...
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If $(e_n)_{n\geq 1}$ is total in $H$ and $\sum \| e_n - f_n \| < 1$, prove $(f_n)$ is total.

Sorry about the title. Not enough space for me. Proposition If $E = (e_n)$ and $F = (f_n)$ are orthonormal sequences in a real Hilbert space $H$ that satisfy $$\sum_{n=1}^\infty \|e_n - f_n\| < ...
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To find an orthonormal basis for the row space of $A$.

To find an orthonormal basis for the row space of $A = \begin{bmatrix} 2 & -1 & -3 \\ -5 & 5 & 3 \\ \end{bmatrix} $. Let $v_1 = (2\ -1 \ -3)$ and $v_2 = (-5 \ \ \ 5 ...
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Normal Vector in the place

I've seen two definitions of a normal vector to a curve in $\mathbb{R^2}$. Suppose we have a parametrisation of our curve: $r(t)=(x(t),y(t)),$ Then differentiating once gives us a tangent vector, ...
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1answer
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Is relationship of orthonormality & orthogonality equivalent to that of squares & rectangles?

To confirm my analogy, I am asking if I can consider every orthonormal basis to be an orthogonal one (but not vv) in the same sense that every square is a rectangle (but not vv). I believe the answer ...
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Are polynomials dense is L2 of the unit disk?

Let $D$ be the unit disk in the complex plane, and let $X$ be the subset of $L^2(D)$ consisting of polynomials in the complex variable $z=x+iy$ with complex coefficients. My question is, is $X$ dense ...
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Showing the multiplication operator only has closed range for characteristic functions

For each $g\in L^\infty$, let $M_g:L^2\rightarrow L^2$ be the multiplication operator defined by $M_g(f)=fg$. Show that the range of $M_g$ is closed if and only if $g$ is a characteristic function. ...
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1answer
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Doubt regarding orthonormal basis

While studying inner product space I came across a theorem that is if V is an inner product space and $\{x_1,x_2,.....x_n\}$ be the orthonormal set of nonzero vectors. Then if $y\in V$ then $ y=\sum_{...
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Find all orthonormal bases of $\mathbb{R^2}$

The question is short: ''Find all orthonormal bases of $\mathbb{R^2}.$ Recall. A basis $\left\{v_1,...,v_n\right\}$ of $V$ is said to be orthogonal if its elements are mutually perpendicular. If in ...
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Merging two orthonormal bases without Gram-Schmidt

I have two sets of column vectors: $A = \{a_1,a_2,\dotsc,a_m\}$ and $B = \{b_1,b_2,\dotsc,b_n\}$. I have orthornormal basis for both of them individiaully. $\{u_1,\dotsc,u_p\}$ is an othornormal basis ...
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Why do we need projection in the definition of the Stokes operator?

$\DeclareMathOperator{\div}{div}$ $\def\bu{\mathbf{u}}$ Let $D$ be the square $[0,1]^2$ and consider the following space: $$ V:=\{\bu: \bu\in H^2(D)^2, \div \bu=0, u|_{\partial D}=0 \}. $$ Introduce ...
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1answer
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Problems with determining the orthonormal basis regarding an inner product $B$

Can somebody tell me where I made a mistake? How would you approach such an exercise? Was my way too complicated? $ B:\mathbb{R}^3\times \mathbb{R}^3\to \mathbb{R},\quad B(x,y)=\sum \limits_{i\neq j}^...
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1answer
35 views

Diagonalizing a real normal matrix

Given the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{pmatrix}$, how would I find a real orthogonal matrix $P$ such that $PAP^t$ is a diagonal matrix? ...
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15 views

Integrate orthogonal function over solid angle

How do I integrate a product of Legendre polynomials over a volume? So I understand that bunch of complete basis orthogonal basis as well. i.e. for Legendre polynomial, $\int_{-1}^1P_n(x)P_m(x)dx=\...
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Why is the maximum value of an orthonormal dictionary of size $N \times N$ less than that of another having size $M \times M$, where M>N?

I was trying to compare the maximum value in DCT dictionary (representing an orthonormal dictionary) of size $N \times N$ to that of another DCT dictionary of size $M \times M$, where $M>N$. I ...
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1answer
23 views

Relation between norm of a vector and sublattice to which it belongs.

Given four rational numbers $m, n, p$, and $q$, define the lattices $L_1 = \{ (m\cdot x, n\cdot x) : x \in \mathbb{Z} \}$ and $L_2 = \{ (p\cdot y, q\cdot y) : y \in \mathbb{Z} \}$. That is, the ...
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1answer
28 views

Does always a $B$ with orthonormal rows/column be found so that $BP=0$?

I am given, $P_{n\times n}$ column stochastic, I need to see whether a matrix of suitable order $B$ with an orthonormal column or rows can be constructed so that $BP=0$? I started trialing like this: ...
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Approximating orthonormal basis in Hilbert space by orthonormal basis from dense subset

Consider a Hilbert space $H_2$ with norm $\|\cdot\|_2$. Consider a linear space $H_1 \subset H_2$ so that $H_1$ is dense in $H_2$. Consider an orthonormal sequence $(h_k)_{k \in \mathbb{N}}$ in $H_2$. ...
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Orthonormal $\{e_n\}$ is complete in $L^2[a,b]$ iff $\sum_{n=1}^{\infty} \mid \int_{a}^{x} e_n(t) dt \mid^2 = x-a$ for all $x \in [a,b]$.

For $n \in \mathbb{N}$, let $\mathcal{E}$=$\{e_n\}$ be an orthonormal sequence in $L^2[a,b]$. We need to show that $\{e_n\}$ is complete in $L^2[a,b]$ iff $\sum_{n=1}^{\infty} \mid \int_{a}^{x} e_n(t) ...
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1answer
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Orthonormal Basis In $l_1$ Norm

If I am working on finite dimensional innerproduct space $\mathbb R^m$, and $\{v_{1},\dots,v_{m}\}$ be a orthonormal basis for $\mathbb R^m$, can I say that they are orthonormal in the sense of $l_1$ ...
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1answer
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Determine orientation of standard orthonormal bases

Let $\vec{e_1},\vec{e_2},\vec{e_3}$ be the standard orthonormal base $(1,0,0),(0,1,0),(0,0,1)$ which is positively oriented. Determine the orientation of each of the following bases: $\vec{e_1},\vec{...
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1answer
81 views

Gram Schmidt Process for a Complex Vector Space

Suppose I have certain independent vectors, say $\lvert V_1\rangle$ and $\lvert V_2\rangle$, which span a 2-dimensional subspace of a given Complex Vector Space on which inner product is defined, how ...
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1answer
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Separable Hilbert Space

Let $\mathcal{H}$ be a separable Hilbert space with complete orthonormal system $e_{1},e_{2},\ldots$ (a) Let $T:\mathcal{H}\rightarrow\mathcal{H}$ be a continuous linear operator. Suppose that there ...
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Sum of Fourier coefficients less than the norm of the respective vector

I have the following question to complete. Let $X$ be an inner product space. Let $(e_{j})_{j\geq1}$ be an orthonormal sequence in $X$. Show that, \begin{align} \sum_{j=1}^{\infty}|(x|e_{j})(y|e_{j})|...
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Calculate an orthonormal basis

We have the following Gram-Schmidt algorithm: I want to calculate for the following vectors an orthonormal basis, with an accuracy of $\epsilon=5\cdot 10^{-3}$. \begin{equation*}a_1=\begin{pmatrix}...
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1answer
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Explicit formula for inner product given orthonormal basis

Suppose $V$ is a $\mathbb C$-vector space and $v_1, v_2, \ldots v_n \in V$ (given explicitly, for instance as vectors with given elements) form its orthonormal basis relative to some inner product $\...
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1answer
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Norm of Fourier series

I am reading the proof of the statement that no non-zero multiplication operator on $L^2([0,1])$ is compact in this post. And I would like to address it as a seperate post as I am only curious about ...
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3answers
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orthonormal basis of infinite dimensional Hilbert space H is not a basis of H as vector space?

Apparently the orthonormal basis $(e_n)_{n\in \mathbb{N}}$ of the Hilbert space $H$ (in special case, infinitly dimensional) is not a basis of $H$ as a vectorspace. Is there a way to prove this?
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Surface integrals, positive or negative normal?

I'm unsure how to decide whether the normal should be positive or negative in $\hat{n}dS=\pm h_2 h_3 e_1 du_2 du_3$, where $h_i$ are the scale factors, $e_i $ are the base vectors, and $u_i$ are the ...
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63 views

Orthogonal projection in Hilbert space

Projector $P\neq0$ is orthogonal projection if and only if satisfied some of the following mutually equivalent conditions: a) $P$ is self-adjoint, $P=P^*$ b) $P$ is normal, i.e. $P^*P=PP^*$ c) $P$ ...
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3answers
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Orthogonal matrices with eigenvalues equally spaced over unit circle?

We know that the eigenvalues of orthogonal matrices have norm 1, and thus they are all on the unit circle. However, I wonder if there is a way to construct a orthogonal matrix (with real entries) ...
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1answer
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“minimization” problem in Euclidean space related to orthonormal basis

I have stumbled upon an exercice for second year undegraduate student majoring in economics which I find quite demanding. I have an idea for the solution, but it seems awfully complicated, and I am ...
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Convergence of sum $f_n(x)=\sum_{l,k} w_{l,n} w_{l,k} x^k$ , with $w$ expansion coefs of an orthonormal system

Good day, Let $\{P_k\}$ be a complete orthonormal system (Fourier series, Legendre-Fourier series, etc..) on interval $(a,b)$ which can be expanded into powers : $$ P_n = \sum_{k=0}^\infty w_{n,k}x^...
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1answer
32 views

A complete orthonormal system $\{e_i\}^\infty_{i=1}$ in $H$ is a basis in $H$

I'm studying about Hilbert space from a book of functional analysis and I just read this theorem (2.1.10) and its' proof. I cannot understand why $(y-x)\perp e_i$? why is it implied?
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12 views

Criteria to find a common non orthonormal basis for two linear operators

I can't find any criteria to determine, in finite dimension, if two operators has a non orthogonal common basis, for example, given two operators A and B if I check $A=A^+$ $B=B^+$ $[A, B] =0$ In ...
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1answer
78 views

Do orthonormal matrices have a geometric interpretation or contextual importance? [closed]

I understand how certain matrices can be orthonormal and the conditions necessary for that. I don't understand its geometric relevance. For instance in a vector space the axis would be orthonormal and ...
2
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2answers
44 views

How can we decompose the identity matrix given a set of orthonormal vectors?

Let $A$ be a positive semidefinite (P.S.D) matrix with distinct set of eigenvalues. since it is P.S.D its eigendecomposition is as follows for eigenpairs of $(\lambda_i,v_i)$ $$ A= \begin{bmatrix} ...
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2answers
37 views

Applying sine orthogonality

I have a confusion as to why this is a viable procedure in the following image: $$ \sum_{n} B_n \int_{0}^{a} \sin \left( \frac{n \pi x}{a}\right) \sin \left( \frac{m \pi x}{a}\right) = \sum_{n} B_n \...
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1answer
63 views

Non total orthonormal set in a non Hilbert inner product space

Suppose there exist a subset $M$ of an inner product space $X$, and the orthogonal complement of $M $ is the zero vector. If $X $ is a Hilbert Space then the span of $M $ will be dense in $X $, but ...