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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Is a linear operator $T$ that is normal, normal regardless of the ordered basis while the matrix representation of that operator must be on an ortho..

Is a linear operator $T$ that is normal, normal regardless of the ordered basis while the matrix representation of that operator must be on an orthonormal basis? I was attempting an exercise that ...
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PCA, properties of norm and or orthonormality of a matrix

I'm reading a paper on a particular form of PCA and I'm struggling to understand one part. We have $||\mathbf{X}-\mathbf{XGH}^T||^2$ with $\mathbf{X}$ a $n\times p$ (data) matrix, $\mathbf{G}$ and $\...
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Matrix of change of basis between orthonormal bases is unitary

Question statement: Let $V$ be a vector space over $\mathbb{C}$ with an inner product $\langle,\rangle$. Assume that $\mathcal{A}=\{w_1, ..., w_n\}$ and $\mathcal{B}=\{v_1,...,v_n\}$ are two ...
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Moments Properties of Orthonormal Basis

The first and second moments properties of an orthonormal basis $\{u_j\}_{j \geq 1}$ are well known: $$ \int_{t\in \mathcal{T}} u_j(t) F(dt)=0 $$ $$ \int_{t\in \mathcal{T}} u^2_j(t) F(dt)=1 $$ and $$ ...
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Inner product space and convergence

We introduce $f ∈ V$ (where $V = C([0,1],\mathbb R)$), and we put $a_n = <f, e_n>$ (where $e_n$ is an orthonormal sequence in $V$). Assuming $g(x) = \sum_{n=0}^∞ a_n e_n(x)$ is continuous, we ...
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Coordinate Vector Relative to Orthogonal Basis

I'm studying for my Linear Algebra final and have came across a problem that I cannot find any examples of in my book and very few online. I wish to know if my solution takes the right steps and ...
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Geometric interpretation of semi-orthonormal matrices

I understand that an orthonormal matrix is normally interpreted as rotation / reflection of the basis vectors since they preserve the angle. But how exactly does this transfer into semi-orthonormal ...
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Inner product space and orthonormality

We have an orthonormal sequence $\{e_n\}$ on the inner product space $V=C([0,\,1],\,\Bbb R)$. Let $h\in V$ and assume $\langle h, e_n\rangle=0$ for integers $n\ge0$. Show $\langle h, p\rangle=0$ for ...
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Is there a more concise way to find the linear combination of a vector with a set of orthonormal vectors?

So I have a set of vectors that I verified are orthonormal. $$ \begin{equation*} A = \begin{pmatrix} \frac{1}{3\sqrt{2}} & \frac{2}{3} & \frac{1}{\sqrt{2}} \\ \frac{1}{3\sqrt{2}} & \frac{...
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Eigenvectors of an Orthogonal Matrix

It is true that for orthogonal matrices, all eigenvectors of a single eigenvalue are orthogonal to all other vectors of different eigenvalues(distinct eigenbases are orthogonal). My question is are ...
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Finding a basis of a subspace which is perpendicular

Let $V = \text{span}\{(1, 1, 0, 1, 1), (1, 2, 0, 2, 1), (1, 3, 3, 3, 1)\}$ be a subspace of $\mathbb R^5$. I found using Gram-Schmidt process that the orthonormal basis of $V$ is $\{(0.5 , 0.5 , 0 , 0....
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Finding the value for this integral

While solving some exercises on orthnormal set on Hilbert spaces, I tried to solve the following problem: Let $\left\{ f_{1},f_{2},...,f_{9}\right\} $ be an orthonormal set in $L^{2} \left[ 0,1\right] ...
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How to compare orthogonal vectors?

I doubt if I am asking this question correctly but for what it’s worth I have a set of orthogonal vectors for which I would like to pick from another set the closest orthogonal vector from it to my ...
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Confused about normalization issue of the orthonormality and completeness relation

Let $V$ be the $\mathbb{C}$-vector space of all mappings from $\{-1, \cdots,-\frac{2}{n},-\frac{1}{n}, 0 ,\frac{1}{n},\frac{2}{n}, \cdots, 1\}$ into $\mathbb{C}^2$. The dimension of $V$ is $2^{2n+1}$. ...
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Can a linear combination of two orthonormal basis vectors always belong to this set?

Consider the set of all unit vectors of the form $ S = (0, x_1, ..., x_n)$. If $\mathbf{e}_1$ and $\mathbf{e}_2$ are members of an orthonormal set, does $\alpha_1 \mathbf{e}_1+ \alpha_2 \mathbf{e}_2$ ...
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The limit of an orthonormal system in $L^2([0,1])$

Given an orthonormal system of functions {$\phi_n$} in $L^2([0,1])$, we let $E$ be the collection of $x \in [0,1]$ such that the limit $\lim_{n \rightarrow \infty} \phi_n(x)$ exists. Define the ...
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Signed Distance of a Point to a Hyperplane

Let's define a affine function $f(x) = \alpha^Tx+\alpha_0$ where $\alpha$ is a vector (weights) and $\alpha_0$ is a constant (bias). Also, define a normal vector of the hyperplane as $n=\frac{\alpha}{...
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Questions about the eigenfunctions and eigenvalues of the momentum operator $\hat{p}$

I'm studying Quantum Mechanics right now and working through an example in the book of an eigenfunction with a continuous spectrum - the momentum operator, $\hat{p} = -i\hbar\frac{d}{dx}$. In the ...
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Axler 6.9: Show that the following list is orthonormal.

Let $n \in \mathbb{Z_{+}}$ and show that the list below is an orthonormal list of vectors in $C[-\pi,\pi]$ in the vector space of real valued functions on $[-\pi,\pi]$ with the inner product given ...
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Absolute value of the determinant of a linear map between two equal dimensional Euclidean vector spaces is independent of orthonormal basis

Let $V_1$ and $V_2$ be two Euclidean vector spaces with the same dimension, and let $f : V_1 \longrightarrow V_2$ be a linear map between them. If $M$ is the matrix of the map $f$ relative to two ...
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If a linear map sends orthonormal basis on orthonotmal basis then it is an isometry?

Let $(\mathbf{u_1,u_2,u_3})$ and $(\mathbf{v_1,v_2,v_3})$ orthonormal lists. Define $T:\mathbb{R^3}\rightarrow \mathbb{R^3}$ through the lineal extension $T(u_k)=v_k$. Is T an isometry? My attempt: I ...
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What am I doing wrong in this Linear Algebra problem?

The question, which is in my Linear Algebra textbook (no answer is provided), goes as follows: Let $W=\text{span}\{v_1= [2,2,-2,0]^T, \hspace{.2cm}v_2=[0, 1, 1, -1]^T, \hspace{.2cm} v_3=[1,1,2,-4]^T\}...
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Can an inner product space contain a complete orthonormal sequence, and still be incomplete?

Can an inner product space contain a complete orthonormal sequence, and still be incomplete? This question came to my mind while reading Optimization by Vector Space Methods by David G. Luenberger. ...
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Determine orthotropic axes of Stiffness Tensor

Consider a fourth oder stiffness tensor $\mathbb{C}$. The components $C_{ijkl}$ are given with respect to a global coordinate system. Is there a way to determine the orthotropic axes from this ...
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Proving linear transformation is normal

I tried to solve the following question to no avail and will appreciate any hints. Let $M \in M_{n×n}(C)$ be an invertible matrix. We'll define the transformation $S : M_{n×n}(C) \to M_{n×n}(C)$ by ...
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Matrix of a linear transformation in an orthonormal basis

Let $B=\{v_1,...,v_n\}$ be an orthonormal basis for $\Bbb R^n$ and $T:R^n \to R^n$ be a linear transformation. Let $A=(a_{ij})$ a representing matrix of the transformation $T$ by basis $B$ (I am sorry ...
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About Finding a Orthogonal Projection

Let $T_A(x)=A\cdot x$, where $$A=\begin{pmatrix}0 & -1 \\ 1 & 0 \end{pmatrix}$$ And the eigenvalues of $T$ are $i$ and $-i$. I have found that $\beta=\{(1,-i),(1,i)\}$ is a orthonormal basis ...
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Is an orthonormal basis the basis for a non-orthonormal basis?

I'm trying to think through the hierarchy of mathematical objects leading to an orthonormal basis. A space may be filled by vectors and a vector may be thought of as a linear combination of ...
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How to efficiently compute Isometry Normal Form and where is an implementation?

I have recently learnt about what was called to me the "Isometry Normal Form", which tells you given an orthogonal operator $P$ on a real inner product space $V$ that you can find an ...
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Proof of Cauchy-Schwarz inequality using a particular result from orthonormal sets

Let {$e_1, ... , e_k$} be an orthonormal set in a finite dimensional real inner product space $V$. If $ v \in V$, I have shown the following result: $\sum_{i=1}^{k} |\langle v,e_i\rangle|^{2} \leq ||...
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Show $Q(x)\cdot Q(y)=x\cdot y\Rightarrow Q^TQ=I$

According to my class notes the following are two equivalent definitions of an orthogonal matrix: $Q^TQ=I$ $Q(x)\cdot Q(y)=x\cdot y$ I've been able to show that $1\Rightarrow 2$, yet I do not know ...
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Do orthonormal changes of basis affect the inner product?

Let $e_1;\ldots,e_n$ and $e_1',\ldots ,e_n'$ be orthonormal basis of a vector space $V$. Consider the vectors $$v=(v_1e_1+\ldots +v_ne_n)=(v_1'e_1'+\ldots +v_n'e_n'),$$ $$u=(u_1e_1+\ldots +u_ne_n)=(...
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What is the relationship between the eigenvalues of $Q^T \Lambda Q$ and $\Lambda$?

Suppose $N\geq p$. Let $Q \in \mathbb{R}^{N \times p}$ has orthonormal columns $q_1,\dots,q_p$. Let $\Lambda \in \mathbb{R}^{N \times N}$ be a diagonal matrix with diagonal entries $\lambda_1 \geq \...
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Find function that minimizes the distance from $f$ to $g$ with respect to the $L_2$-norm

Let H be the Hilbert space $L_2([-1,1])$ with the standard inner product: $$\langle f|g \rangle=\int_{-1}^1 \bar{f}(x)g(x)dx$$ and define the functions, $f_n$, $n=0,1,2,3,...,$ on $[-1,1]$ by $$f_n(x)=...
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Frobenius norm equality related to "orthonormal pair"

Related to this question as far as concerned the paper discussed. In the following theorem the author is not very clear about who $(W,W-)$ and $M$ should be. Morover, the notation orthonormal pair I'm ...
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Proving that matrices in $O(2)$ are of one of two forms

Given $A \in O(2)$, show that $A$ has one of the following two forms: $$ A = \left( \begin{matrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{matrix} \right) $$ $$ A = \...
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Relationship between angle of vectors and orthogonalization

I'm trying to understand the following question: Let $u_1,...,u_d$ be a set of orthonormal vectors in $\mathbb{R}^n$. Let $a,b$ be unit vectors in $\mathbb{R}^n$ such that $\angle(a,b) < \epsilon$. ...
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Operator norm in Hilbert space, Schur criterion for infnite matrices

Let H be Hilbert with an orthonormal basis $(e_n)_n$. Consider $A,B>0$ and two sequence $a_n>0, b_n>0$ such that $$\sum_{i=1}^\infty b_i(T e_i, e_j)\leq Aa_j \quad\text{and}\quad \sum_{j=1}^\...
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Finding vectors orthonormal to a given vector set and the Gram-Schmidt process

Here is a likely very simple problem that I am confused about: Let $\{v_k\}$ be a set of vectors ($k=1, ..., n$). I would like to find a set of vectors $\{q_k\}$ such that $$\langle q_i| v_j \rangle = ...
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Orthonormal Vectors with Respect to an Inner Product

I am given an inner product defined as the following image: Inner product I have already found the range of the values of α and β. For the following question, I do not know how to approach the problem....
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Does the best vector approximation in a subspace require a normalized basis?

In Linear Algebra by Hoffman and Kunze, on page 284, they say Theorem 4. Let $W$ be a subspace of an inner product space $V$ and let $\beta$ be a vector in $V$. ... (iii) If $W$ is finite-dimensional ...
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Proof that bases can be extended while maintaining orthonormality

If we have a subspace $U$ of $V$, where $U$ has an orthonormal basis $(u_1, u_2, \dots, u_n)$, how do we show that we can extend the basis to $(u_1,u_2,\dots,u_n,v_1,v_2,\dots,v_r)$ such that it is ...
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Every incomplete inner product space has a maximal but incomplete orthonormal system

A functional analysis text book contained the following exercise. Let $V$ be an incomplete inner product space. Show there exists a maximal orthonormal system that is not complete. Hint: Use a closed ...
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Hodge star and non-orthonormal basis

I am working with the Hodge star and a non-orthonormal basis, and I can't see where exactly something is going wrong. Let $V$ be an oriented $n$-dimensional vector space, and let $\Lambda V$ be the ...
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Components of vector as an inner product

I am making my way through the book Mathematical methods for physics and engineering and am stuck trying to understand orthonormal bases. So far, I have been introduced to the idea of an N-dimensional ...
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Are Fourier series a basis for $L^2((-\pi,\pi)^m)$?

It is a classical result that Fourier series are a basis for $L^2((-\pi,\pi))$. The proof can be found in the Rudin and uses isometry properties of the Fourier transform. Now, I am struggling to see ...
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Orthonormal basis of matrices

I practice some exercises in linear algebra and suddendly I have to compute a orthonormal basis for the subspace $\mathbb{M}_{2,2}$ of the following matrices given below. $$V_1=\begin{bmatrix} 1 & ...
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Prove $\sqrt2\sin(\pi pj\Delta x)$ is ONB of $\{U \in \mathbb{R}^{M+1} \mid U_1=U_M=0\}$ w.r.t. $(U,W)=\Delta x\sum_{j=0}^{M}U_jW_j$

Here is the full exercise: From my understanding I need to show: $||\varphi_p||^2=\frac{1}{\Delta x}\sum_{j=0}^{M}\sin^2(\pi pj\Delta x)=1 $ $p \neq q \implies (\varphi_p,\varphi_q)=0$ $\...
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Proving that a bounded linear operator $A \in \mathscr{I}_1$

Can someone please help me with the following problem? I have some of my work below but I am not sure if I attacked the problem wisely. Thank you for your time and consideration. Suppose that a ...
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Questions regarding answer.

Consider this post. Questions: For equation (2) from Abel, did the person mean to say $$b - \alpha q_1 - \color{red}\beta q_2 = (q_1^Tb - \alpha)q_1 + (q_2^Tb-\beta)q_2 + \epsilon?$$ Working out the ...
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