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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Proof about the orthonormality of a set of integer translates.

I am struggling with the following proof. Prove that if $\psi\in L^2(\mathbb{R})$ then $\{\psi(t-n)\}_{n=-\infty}^{\infty}$ is an orthonormal set if and only if $\Psi(\xi)=\sum_{n=-\infty}^{\infty}|\...
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Relation between squared norms and sets of orthonormal vectors having the same span

I was asked to show the following claim but I'm stuck. It seems I can't find the right reasoning path. Given a matrix $M\in\mathbb{R}^{n\times d}$ and two sets of pairwise orthogonal unit vectors {$...
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Best approximation of a function among closed linear manifolds

Let $H$ be an infinite-dimensional Hilbert space and consider a $n-dimensional$ closed linear manifold generated by a subset of orthonormal basis, say, $M = span(\{u_1,u_2,\cdots,u_n\})$. Of course, ...
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Show that each vector in an n-dimensional vector space can be represented as the summation of its components along the orthonormal basis.

Show that in an n-dimensional vector space V over the universal set with orthogonal basis {$a_1, a_2,..., a_n$}, each vector B can be expressed as: B = $\frac{<B,a_1>a_1}{||a_1||^2}$ + $\frac{...
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Can the proof about direct sum decomposition of the inner product space be generalize to infinitely dimension space

There is a theorem about the finite-dimensional inner product space. Suppose a finite-dimensional inner product space $V$ with a subspace $W$, then $V=W\bigoplus W^{\bot}$. And the proof is as ...
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In the case below, calculate (i) the orthogonal projection $ P_W $ associated with the two-dimensional vector subspace W = L (S ') of the …

In the case below, calculate (i) the orthogonal projection $ P_W $ associated with the two-dimensional vector subspace W = L (S ') of the vector space V generated by linearly independent set S '= $ \{\...
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Finding constant for orthonormality

Assume the problem \begin{align} f''(x) &= \lambda f(x) \\ f'(0) &= f(1) = 0 \\ \end{align} with solution: $$ \phi_n(x) = c \cdot\cos\left( \frac{(2n-1)\pi}{2}x\right), \quad n = 1,2,\dots $...
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Question about Orthonormal Bases

I solved this question but I am not quite sure if it is a valid solution. The question is: Let $u$ be a vector in an inner product space $V$ and let $\{v_1, v_2, v_3, \ldots, v_n\}$ be an ...
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Calculate the matrix A of the linear transformation T: V→ V in the base S, the matrix C of change from the orthonormal base S to the orthonormal …

Calculate in the cases below (i) the matrix $A $ of the linear transformation $T: V \rightarrow V$ in the base $S \subset V$ and (ii) the matrix $C$ of change from the orthonormal base $S$ to the ...
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Hermitian/Skew Hermitian Matrix for non-orthogonal basis

Assume that the matrix $A$ is Hermitian/skew-Hermitian. There is a theorem, which says that if this matrix represents some transformation $T$ with respect to an orthonormal basis, then this ...
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43 views

Orthonormal basis of $L^2(0,1)$

Show that the only functions that satisfy $f''(\xi) = \lambda f(\xi)$, $f'(0) = f(1) = 0$, $\|f\|_2 = 1$ for some $\lambda \in \mathbb{R}$ are the functions: $$ \phi_n(\xi) = \sqrt{2}\cos\...
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number of elements of a basis of a subspace of R4

If I've got $ U $ a subspace of $\mathbb{R}^4$ $$U = < (1,-1,0,0),(0,1,1,1),(2,1,0,1) >$$ And I want to find an orthonormal base for the subspace $U$ My doubt is: Can I make an orthonormal ...
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Adjoint of right shift operator on orthonormal basis $(e_{n})_{n\in\mathbb{N}}$ of $\ell^{2}(\mathbb{N})$

I'm sorry if this question is a duplicate. Suppose $(e_{n})_{n\in\mathbb{N}}$ is the usual orthonormal basis of $\ell^{2}(\mathbb{N})$. We can define an operator $v\colon H\to H$ by $ve_{n}:=e_{n+1}$....
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Prove: $\|x\|^2=\sum_{i=1}^k|\langle x,e_i\rangle|^2\iff x\in\operatorname{span}\{e_1,\ldots,e_k\}$

Let $\{e_1,\ldots,e_k\}$ be an orthonormal set in a unitary space $V$. Prove: $$\|x\|^2=\sum\limits_{i=1}^k|\langle x,e_i\rangle|^2\iff x\in\operatorname{span}\{e_1,\ldots,e_k\}$$ My attempt: My ...
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Orthonormal sequence and Gram schmidt

Let $V$ be a Inner product space, $(\epsilon_1....\epsilon_k)$ orthonormal sequence, let $\vec{v_1},\vec{v_2}\in V$. $\quad$ $GS(u_1...u_k)$ is the set of the vectors that recieved after doing GS ...
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Orthonormal basis with parameters

I have the set : $$W=[(x_1,x_2,x_3,x_4)\in{\mathbb{R}^4}\quad| \quad x_1-x_2-x_3-x_4=0]$$ I need to find orthonormal basis, So I found just one vector and I have only parameters in my answer and I am ...
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27 views

Characterization of orthonomal maximal systems in a Hilbert space

Let $H$ a Hilbert space and $B=\{e_\alpha\}_{\alpha\in\Lambda}$ a orthonormal system in H. Prove $B$ is orthonormal maximal in $H$ iff for each $x\in H$ such that $\langle x,e_\alpha\rangle=0\quad\...
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Trace norm of rank one operator $x\otimes y$ for $x,y\in H$

Let $H$ be a Hilbert space. The trace norm on $B(H)$ is defined as $$\|u\|_{1}:=\operatorname{tr}(|u|):=\sum_{e\in E}\langle|u|(e),e\rangle,$$ where $|u|:=(u^{*}u)^{1/2}$ and $E$ is (any) orthonormal ...
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Rotation matrix $R$ on orthonormalbasis of $\mathbb{R}^2$ [closed]

I'm working on this problem and have no clue how to solve it. Any idea, hint or solution would be very appreciated. Let $E$=$\mathbb{R}^2$, equipped with the standard scalar product. Let $R_\theta$ ...
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1answer
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Orthonormal matrix whose column space is orthogonal to another matrix.

I am reading a paper on controlling false discovery rate for variable selection. This paper constructs Knockoff matrix $\tilde{X}$ for original design $X$ using equation $$ \tilde{X} = X\left ( ...
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Prove T is an isometry if and only if ${T(e_1),T(e_2),\cdots T(e_n)}$ forms an orthonormal basis of $R^n$

Suppose the linear transformation $T:R^n \rightarrow R^n$ where T(x)=Ax. Prove that T is an isometry if and only if ${T(e_1),T(e_2),\cdots T(e_n)}$ forms an orthonormal basis of R^n. I started ...
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Determine eigenvalues and eigenvectors for 2D flowfield functies

I got an assignment and I am stuck determining whats asked. A velocity field and and additional functions are given and I am not sure how to use these, hopefully someone can help me out. The ...
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What is adjoint of extended right shift operator on a Hilbert space?

If $H$ is an infinite dimensional (not necessarily separable) Hilbert space and $(e_{n})$ is an orthonormal sequence. One can define $T(e_{n}):=e_{n+1}$, extend linearly to $\operatorname{span}\{e_{n}\...
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Orthonormalization of basis - Diagonalization of overlap matrix.

Is it sufficient to diagonalize overlap matrix (Gram matrix) in order to find orthogonalized basis? Overlap matrix $G$ can be diagonalized as $P^{-1} G P=\lambda$. My new set of vectors would be $\...
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1answer
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How to show that the row vectors of a special unitary group of degree 2 form an orthonormal basis?

I'm studying about special unitary groups and came across a problem that I'm having trouble with. If we define a special unitary group of degree $2$ as: $$\text{SU}(2) = \left\{ \begin{bmatrix}...
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Inner product and orthogonality in non-orthonormal basis

Suppose that $V$ is an inner product space and that $\mathbb{K}$ is a field. The inner product is a map $\langle \cdot,\cdot \rangle : V \times V \to \mathbb{K}$. In the Euclidean space $\mathbb{R}^n$ ...
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Orthonormal bases and the spectral theorem?

Let $m, n \in N$. Prove that for any linear transformation $T: R^n \rightarrow R^m$ there exists an orthonormal basis $U = (\vec{u_1},...,\vec{u_n})$ of $R^n$ such that for all $1 \leq i,j \leq n$, ...
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1answer
47 views

Equivalence and equality of a matrix and its diagonal matrix

I have the following task: Let $A = \left( \begin{array}{ccc} 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 \end{array} \right)$. ...
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1answer
28 views

inverse SVD square matrix with rectangular orthogonal matrix

I'm trying to prove the inverse matrix of $X=U\Sigma^{}V^T$ is $X^{-1}=V\Sigma^{-1}U^T$. X is invertible and dxd square matrix, U and V are orthogonal matrices, which are dxk, and $\Sigma$ is kxk ...
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Finding orthogonal vector to orthonormal basis vectors [closed]

If you are given an orthonormal basis vector set in a defined space are you still able to find an additional vector that is orthogonal to all the vectors included in the basis? (Assuming a non-zero ...
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If $H$ is Hilbert and $T\in B(H)$ is left shift w.r.t. the OB $(e_{k})_{k\in\mathbb{N}}$, then $T^{n}x\to0$ for all $x\in H$

Let $H$ be Hilbert and $T\in B(H)$ the left shift operator w.r.t. the orthonormal basis $(e_{k})_{k\in\mathbb{N}}$. That is, $Te_{1}=0$ and $Te_{k}=e_{k-1}$ for $k>1$. Then how do I show that $T^{n}...
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Orthonormal Basis of $\mathbb{R}^4$

I'm working on a problem that states: Find an orthonormal Basis of $\mathbb{R}^4$ with respect to the bilinear form defined through $$P= \begin{pmatrix}1&-2&1&-1\\-2&13&-6&...
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Question regarding the characteristics of Orthogonal matrices

Let us assume that matrices Q1 and Q2 are orthogonal matrices. I have some questions regarding their characteristics. Case 1: Case 1 question In this case, I thought that this relationship would be ...
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Proving $(e_{j})(k)=\frac{1}{\sqrt{N}}e^{(2\pi ijk)/N}$ is an orthonormal basis for $\mathbb C^n$

I know that in order to do this I need to show that $\langle e_j,e_l \rangle =\delta_{j,l}$ and I can show the fact that if $j=l$ I get $1$ but I'm really struggling showing $\langle e_j,e_l\rangle =0$...
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Scalar Product on $\mathbb{R}^3$

I have a question on how to solve the following problem. Find a scalar product on $V=\mathbb{R}^3$ such that $\{e_1, e_2, e_3\}$ defined as $e_1=(1,1,0)$, $e_2=(1,0,1)$, $e_3=(0,1,1)$ is an ...
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1answer
25 views

Entries of a unitary matrix

In the solution a a problem in quantum computation I saw this line: $$U_{ij}=\langle\psi_i|\left(\sum_k|\phi_k\rangle\!\langle\psi_k|\right) |\psi_j\rangle.$$ Where $U_{ij}$ are the entries of a ...
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If $U^TU$ is the identity matrix, then the columns of $U$ form an orthonormal set?

I have the following statement: "If $U^TU$ is the identity matrix, then the columns of $U$ form an orthonormal set". I want to figure out if it is true or not. By the way I know that it works with ...
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Dot product / inner product / orthonormal basis

Dot product is "just" a specific example of an inner product. Is this right? I am asked to find a scalar product (I think they mean the inner product) on $$ V = \mathbb{R}^3 s.t. \left( \begin{...
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Use the Gram-Schmidt process to find the orthonormal basis for the row space of the matrix $A$.

Use the Gram-Schmidt process to find the orthonormal basis for the row space of the matrix $A$. The matrix $A$ is as follows: \begin{bmatrix}1&1&0&0\\-1&3&0&1\\-3&1&-...
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Show that a finite set of matrices is an orthonormal system

Assume that $$S_1 = \begin{pmatrix}1&0&1\\ 0&1&0\\ 1&0&0\end{pmatrix}, S_2 = \begin{pmatrix}1&0&1\\ 0&-3&0\\ 1&0&0\end{pmatrix}, S_3 = \begin{pmatrix}0&...
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48 views

Show that a finite set $B = \left\{1, x, x^2 \right\}$ is an orthonormal system

Show that a finite set $B = \left\{1, x, x^2 \right\}$ is an orthonormal system with respect to the inner product $$\left \langle f, g \right \rangle = \int _{-1}^1\:f\left(t\right)\cdot g\left(t\...
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Showing estimate in Hilbert space

I've been puzzling with a math problem for a while and hope you can help me. There is given a Hilbert space $H$ with an orthonormal basis $(v_i)_{i\in \mathbb{N}}$ and an increasing sequence $(\...
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42 views

Find the orthonormal basis for the subspace $U$ of $M_{2,2}(\mathbb{R})$ spanned by

Consider the real inner product space $M_{2,2}(\mathbb{R})$ (the space of 2 x 2 matrices with real entries), with inner product: (a) Find the orthonormal basis for the subspace $U$ of $M_{2,2}(\...
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1answer
74 views

Dirac functions, inner products and $T \in \mathcal{L}(G)$

If G is a countable group with neutral element e (and with the composition written multiplicatively). $\ell^2(G)$ consist of functions $x: G \to \mathbb{C} $ such that $\sum_{t \in G} \vert x(t) \...
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algebra - scalar product and ortonormal bases

I've got an ALGEBRA question. We've got the scalar product · defined in R3 and the base B={(1,0,-1),(1,-1,-1),(0,1,1)} is orthonormal to that scalar product. We have to calculate the general ...
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Prove: If a linear map $L$ has a symmetric matrix w.r.t. one orthonormal basis, then it has a symmetric matrix w.r.t. all orthonormal bases.

My attempt: Suppose that $\{e_1,\dots,e_n\}$ is an orthonormal basis and $L(e_i)=a_{ik}e_k$ (Einstein sommation). Then, $(a_{ik})$ is a symmetric matrix (by given information). Now consider another ...
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1answer
32 views

simple Application of Gram-Schmidt Orthogonalization

I wanted to apply the Gram-Schmidt orthogonalization to the following simple example of polynomials $1,x,x^2,x^3$ in $L^2[-1,1]$, is it correct? $e_1 = 1$ $e'_2 = x - \int_{-1}^{1}xdx = x \implies ...
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1answer
39 views

Find an orthonormal basis for the subspace…

Consider the real inner product space $M_{2,2}$ $\mathbb{R}$ (The space of 2x2 matrices with real entries), with the inner product as follows: (a) Find an orthonormal basis for the subspace U of $M_{...
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2answers
101 views

How to impose orthonormality constraints by method of Lagrange multipliers

I want to find the matrix $\Phi: \mathbb{R}^n \to \mathbb{R}^m$, $m<n$ that minimizes $$V={\rm tr}(\Phi R \Phi^T)$$ subject to the orthonormality constraint $$\Phi\Phi^T=I$$ where $R: \mathbb{R}^n \...
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55 views

Is the Gram-Schmidt procedure a bijection?

Since the Gram-Schmidt procedure yields a single orthonormal output for a given linearly independent input, we can view the procedure as a function which maps an original linearly independent list of ...

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