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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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Show that $|\det (a_1, \ldots, a_k, b_1, \ldots, b_{n-k})|=\left|\det\left(\left\langle a_i, d_j\right\rangle \right)\right|$

Let $U$, $L$ be subspaces such that $U \oplus L= \mathbb{R}^n$, and choose orthonormal bases $a_1,\dots,a_k$ for $U$, $b_1,\dots,b_{n-k}$ for $L$, and $d_1,\dots, d_{k}$ for $L^\bot$. I want to show ...
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Concept of signal size for energy saving in optimization

I've been trying to redo this optimization problem from this paper, but on GEKKO Python code instead of MatLAB as they did, which is about finding the maximum Biodiesel concentration at final time: J =...
Luiz Miguel's user avatar
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Orthonormal basis for complex vector space

I have to answer the question with true or false: Every complex vector space with an inner dot product has an orthonormal basis. I think it is false for the case $\dim V= \infty$. But i cant find a ...
wertz1212's user avatar
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Why the orthogonal projection give minimal?

Given the data $ \vec{x}=(-2,-1,1,2) )$ and $( y=(1,1,-1,1) )$. Use an orthogonal projection to determine the coefficients $( a_{0}, a_{1}, a_{2} )$ of the quadratic polynomial function $ \begin{...
asdfgh jkl's user avatar
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Find a matrix $Q$ that has $Q(Q^T)=I$ but does not have orthonormal columns

My textbook says that if a matrix $Q$ is square and has orthonormal columns then $Q(Q^T)=I$, but it does not say the opposite (that if $Q(Q^T)=I$ then $Q$ has orthonormal columns). Is there an example ...
Bob Joe's user avatar
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$V$ Euclidean iff there exist linear isometry that $f(v)=w$ for $|v|=|w|$

From the book 'a course in metric geometry' exercise 1.2.24 : Let $V$ be a finite-dimensional normed space. prove that V is Euclidean iff for any tow vectors $v,w\in V$ such that $|v|=|w|$ there ...
hr1380's user avatar
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If $\sum_{n=1}^\infty a_n^2$ converges, then $\lim_{n\to \infty} \frac 1 {\sqrt n} \sum_{k=1}^n a_k =0$

Problem. Prove that if $\sum_{n=1}^\infty a_n^2$ converges, then $\lim_{n\to \infty} \frac 1 {\sqrt n} \sum_{k=1}^n a_k =0$. The problem arises from the following question: Let $(e_i)_{i=1}^\infty $ ...
Robert's user avatar
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Is here a mistake or can you explain me what orthonormal basis mean?

I am currently trying to get insight into SVD, and I found one book with an explanation of how we find the 𝑉 and 𝑈 matrices and why it holds that any 𝑚×𝑛 matrix can be represented in this ...
comediann's user avatar
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Projection of vectors from starting basis onto orthogonal complement

Gram-Schmidt process allows us to produce a basis $\{w_1,...,w_n\}$ starting from a basis $\{v_1,...,v_n\}$. If I define $W_j$ to be the subspace generated by $\{w_1,...,w_j\}$ for $j=1,..,n.$ Can I ...
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How to find the tangent when converting from 2D angles to Spherical to Cartesian coordinates?

I am making a spherical/ball-in-socket joint, and I want to limit the movement of the bodies relative to each other. The limit is defined as 2 angles $\alpha$ and $\beta$ which make a 2D rectangle. I ...
Liburia's user avatar
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Euclidean norm of a vector resulting from a matrix multiplied by an orthonormal matrix multiplied by an arbitrary vector!

Let $A \in \{0,1\}^{m\times n}$ be a binary matrix with $ m < n$, and let $U \in \mathbb{C}^{n \times n}$ be an arbitrary orthonormal matrix. Let $\sigma_{min}$ denote the smallest non-zero ...
Drimitive Watson's user avatar
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Is there another way of finding the eigenvectors?

In the following exercise I am asked to find the orthogonal matrix $P$ such that $P^tFP$ is in normal form (diagonal?). Where $$F=\frac{1}{4}\begin{bmatrix}\sqrt{3} & \sqrt{3} & 3 & -1 \\\ ...
MSU's user avatar
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Bilinear forms, endomorphisms and skew symmetric matrices

Let $(V, g)$ be an inner product space and $f$ and endomorphism of $V$ such that $$ g(u,f(v)) = -g(f(u),v),\ \ \text{for all}\ u,v \in V. $$ Prove that a) $\ker(f)$ and $Im (f)$ are orthogonal ...
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Why does this expression for an orthogonal projector work?

With the normal matrix $F$, I am asked to find the orthogonal projectors onto each of its subspaces. In my class notes I found out the following reasoning: $$P_i \mathbf{u}_i=\alpha \mathbf{u}_i$$ ...
MSU's user avatar
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Why did my teacher solved this problem this way?

The problem asks the following: which vector of the subspace $$V= \{\mathbf{x} \in \Bbb{R}^4: 2x_1+x_2+x_3+3x_4=0; 3x_1+2x_2+2x_3+x_4=0; x_1+2x_2+2x_3-9x_4=0\}$$ gives the best approximation to $(7,-4,...
MSU's user avatar
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Orthonormal system build upon a Fourier Transform

I'm currently taking a Fourier Transform course and I have to solve the next problem: "Suppose you have a function $f$, continuous in $[0,1]$ and with compact support also $[0,1]$ (that is, $f(x)=...
Pedro Mateo piqueras's user avatar
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1 answer
148 views

Is union of orthonormal bases orthonormal?

Let a matrix the $A \in M_{n\times n}(\mathbb{R})$, and has set of eigenvalues, $\sigma(A)$={$\lambda_1$,$\lambda_2$........,$\lambda_k$}, that is $\forall \lambda \in \sigma(A)$ such that orthonormal ...
user avatar
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1 answer
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Is the nullspace of transpose of any matrix orthogonal to the range of that matrix?

Let a matrix the $A \in M_{n\times n}(\mathbb{R})$. My question is why every matrix $A$ satisfies $R(A) \perp N(A^T)$(where $R(A),N(A^T)$ are range of $A$,null space of $A^T$ respectively)? In ...
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Inequality involving a symmetric matrix and minors of an orthogonal matrix

Fix $n \geq 3$ and take any orthonormal vectors $x,y,z \in \mathbb{R}^n$. Let also $A \in M_n(\mathbb{R})$ be a symmetric matrix with positive entries ($A_{ij} = A_{ji} > 0$). Is the following ...
meler's user avatar
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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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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 ...
CountDOOKU's user avatar
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For the following inner product space $V$ and $T ∈ L(V )$, evaluate$ T ^ ∗$ at a given point in $V$

Pardon for the loose title but I have this question: For the following inner product space $V$ and $T ∈ L(V )$, evaluate $T^*$ at a given point in $ V$: $V=P_1(R)$, with $\langle f(x),g(x) \rangle =\...
Kshitij Kumar's user avatar
2 votes
0 answers
40 views

basis in $H^{-1}(0,1)$

It is known that $(\sin(n\pi x))_{n\ge 1}$ is an orthogonal basis in $L^2(0,1)$. My question is whether it is also a basis for $H^{-1}(0,1)$, the dual space of $H^1_0(0,1)$? I guess it is not, but I'm ...
Migalobe's user avatar
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3 votes
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Why does the Haar orthonormal system span the whole $L^2$?

I am reading "Real Analysis with an Introduction to Wavelets and Applications" because I want to understand wavelets better for my work. I got stuck on a detail about the Haar Basis in ...
Matteo Aldovardi's user avatar
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31 views

Extend an orthogonal set of vectors to an orthonormal basis in SVD.

I'm learning about Singular Value Decomposition (SVD) and how to compute each matrix in the decomposition $A=U\Sigma V^T$. I know how to compute $\Sigma$ and $V$ and hence most of $U$, since we know ...
HBH's user avatar
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Does there exist an orthonomormal basis of $L^2[-\pi,\pi]$

Define $L^2[-\pi,\pi]$={ $f:[-\pi,\pi] \to \Bbb R$ or $\Bbb C$, and $f$ is square integrable }, where I want to consider real and complex value measurable functions separetely.I know that there exist ...
lee's user avatar
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lengths of orthogonal projections of the standard basis on a subspace

Let $e_1,\ldots,e_n$ be the standard basis in $\mathbb{R}^n$. Suppose the scalars $\lambda_1,\ldots,\lambda_n$ satisfy $0< \lambda_1,\ldots,\lambda_n\leq1$ and $\lambda_1^2+\ldots+\lambda_n^2 = m$, ...
Ayden Chang's user avatar
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0 answers
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Orthonormal basis of the Fourier Transform

I am currently writing a small paper regarding the analogy between Fourier analysis and linear algebra for my mathematics class. Unfortunately, I am stuck at proving that $\mathrm{e}^{2\pi\mathrm{t}\...
Fynn Zentner's user avatar
2 votes
1 answer
88 views

Confusion regarding orthonormal basis of $L^2[0, 1]$, in requiring $f(0) = f(1)$?

Let us consider here the continuous elements of $L^2$. It is often stated that the family $e_k(x) = e^{-2 \pi i k x}$ is an orthonormal basis of $L^2[0, 1]$, in that a function can be written as $$ f(...
Drew Brady'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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5 votes
2 answers
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How is it possible for the L² norm of f − g to measure the area between the graphs of f and g?

Here is the definition of a norm given by my textbook; (This is from Fourier Series and Boundary Value Problems by James Ward Brown and Ruel V. Churchill, Chapter 7) I'm confused by what authors say ...
ant's user avatar
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If $(v_n)_n$ is a Hilbert basis for $H$ over $\mathbb{C}$, can you turn $(\mathrm{Re}(v_n), \mathrm{Im}(v_n))_n$ also to a Hilbert basis for $H$?

(Question:) Suppose that $H$ is a Hilbert space over the field $\mathbb{C}$ and that $(v_n)_{n=1}^\infty$ is a Hilbert basis for it, that is a sequence of orthonormal vectors whose span is dense in $H$...
Epsilon Away's user avatar
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Completes of an orthonormal set in a Hilbert space

Let numbers $\{\rho_n\}_{n\ge 0}, \quad \rho_n \neq \rho_k,\quad (n\neq k)\quad$ of the form $\quad \rho_n = n + \frac{a}{n}+\frac{\kappa_n}{n}, \quad \{\kappa_n\} \in \ell^2 \quad $ be given. Then $$ ...
Juan's user avatar
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1 answer
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Norm of cos in $L^2[-T,T]$

I am reading the book Applied Fourier Analysis by Tim Olson. There I got introduced the Fourier Series and we have derived the coefficients $a_k$ and $b_k$ for the Fourier Series on $L^2[-\pi,\pi]$. ...
Thomas Christopher Davies's user avatar
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1 answer
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Irrational Number Decomposition

We know that for a function (adhering to certain rules). we can have a Fourier series, which gives us the function as the sum of sines and cosines. Thus, we can say that almost all functions (adhering ...
Suyash's user avatar
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1 answer
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Trouble verifying a step in a proof regarding the span of an orthonormal family

I'm reading Sheldon Axler's Measure, Integration & Real Analysis and I am confused about a step that the reader is asked to verify. The theorem is: Suppose $\left\{e_k\right\}_{k \in \Gamma}$ is ...
ctk's user avatar
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Change of basis and transformation of a vector.

I am given the matrix $A=\begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}$, and a vector $x=\begin{bmatrix} 1 \\ 1 \end{bmatrix}.$ After computing the eigenvalues and normalized eigenvectors, we ...
Akis's user avatar
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Question about derivation of Fourier transform

I'm reading the book "Fourier Analysis and Its Applications" and in the derivation of the Fourier transform he began with writing the Fourier series $$f(t)=\sum_{n=-\infty }^{\infty }c_{n}...
Mans's user avatar
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1 vote
3 answers
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Given $\|x_1\|=\|x_2\|$, then there is an orthogonal matrix $\Gamma$ such that $x_2=\Gamma x_1$.

Let $x_1, x_2$ be members of $\mathbb R^p$ such that $\|x_1\|=\|x_2\|$. Then there is an orthogonal matrix $\Gamma$ such that $x_2=\Gamma x_1$. How to prove the above? I know given $x_2=\Gamma x_1$, ...
Jonathen's user avatar
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How is this stability bound for the unique interpolant possible? $|s^*(x)|^2\le K(x,x)\|f\|_K\text{cond}_2(G_S)$

If one wants to reconstruct a function $f$ which, we assume is an element of a Hilbertspace $(\mathcal{H}(\Omega,K),(\cdot,\cdot)_K)$ of functions $\Omega\to\mathbb{R}$ with a reproducing Kernel (a.k....
Max Stuthmann's user avatar
1 vote
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Basis of the rows space after multiplication with an orthonormal matrix.

Given a matrix $A \in \mathbb{R}^{n\times m}$. Given another column orthonormal matrix $R \in \mathbb{R}^{n\times \ell}$. Now wo do dimensionality reduction of matrix $A$ using $R$ as follows: $$B = R^...
Zenator 's user avatar
-4 votes
1 answer
103 views

Striving towards a more precise, truthful, and accurate understanding of why gimbal lock happens. [closed]

Please, somebody explain why gimbal lock happens properly. Not a single correct solution out there, people always seem to gloss over the important details and if you dig into the common explanations, ...
Dude's user avatar
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Brezis' exercise 8.28.10: prove that there exists a constant $a \in \mathbb{R}$ (depending on $f)$ such that $(k \alpha_k(f)+a)_k \in \ell^2$

Let $I$ be the open interval $(0, 1)$ and $H := L^2 (I)$ equipped with the usual inner product $\langle \cdot, \cdot \rangle$. Consider the linear map $T: H \to H$ defined by $$ (Tf) (x) = \int_0^x t ...
Akira's user avatar
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Orthonormal Basis of $L^2[0,1]$ having indicator functions?

I was asked to construct an ONB of $L^2[0,1]$ having functions taking at most two values. By suitably scaling, I can think them as indicator functions. So question boils down to finding countably many ...
Anirban Sarkar's user avatar
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1 answer
45 views

Orthogonal Bases and Fourier Approximation

I am confused about how my professor resolved a fourier approximation. The question was: Find the fourth-order fourier approximation of: $$f(x) = \sin (5x)$$ on the interval: $$0\le x \le 2\pi$$ He ...
MattKuehr's user avatar
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The converse for the basis of tensor product of Hilbert spaces

I have been studying the tensor product of Hilbert spaces, and I know that if ${\psi_k}$ and ${\phi_l}$ are orthonormal basis of $\mathcal{H}_1$ and $\mathcal{H}_2$ respectively, then ${\psi_k\otimes\...
Thomas Belichick's user avatar
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98 views

Absolute value of dot product of high-dimensional vector

My professor during the class mentioned (page 35) that for two vectors in high-dimensional space (say dimension $d$), we typically have $$ \left|\boldsymbol{u}^T \boldsymbol{v}\right| \approx \frac{\|\...
Chris XU's user avatar
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1 answer
82 views

Riesz Representation Theorem & Weak closure of orthonormal basis in Hilbert space [closed]

So here it was shown in an infinite dimensional seperable Hilbert space $H$, with the orthonormal basis $E=\{e_1,e_2,\dots \}$ that $0\in \overline{E}^w$, where the right hand side denotes the weak ...
MarvinsSister's user avatar
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1 answer
72 views

Prove that which is an orthonomal basis of $L^2(R)$

Prove $$\left \{\frac{1}{π^{\frac{1}{2}}}\left (\frac{i-x}{i+x} \right )^n\frac{1}{i+x} \right \}_{n=-\infty }^{n=\infty}$$is an orthonomal basis of $L^2(R)$ I tried something like taking the inner ...
tianhaowu's user avatar
1 vote
0 answers
87 views

Representation of closed linear span of an orthonormal set in Hilbert space

This is a follow-up of this post. Here is the statement we want to prove. Let $\{x_j\}$ be an arbitrarily indexed orthonormal set in a Hilbert space(possibly non-separable). Show that the closed ...
sum_math's user avatar
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