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Questions tagged [orthogonality]

This tag can be used to refer to the orthogonality of a set of vectors, a matrix or a linear transformation.

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Almost orthogonal operators after a relative scaling

If two positive operators $Q_1$ and $Q_2$ with unit $\ell_1$ norm are almost orthogonal: $\parallel Q_1 - Q_2 \parallel_1 \geq 2 -\epsilon$, then what can we say about the operators $Q_1$ and $c Q_2$, ...
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How to find the generator matrix for $C/C^{\perp}$?

Background/my workings: I am reading a paper which talks about the $[6,5,2]$ classical binary single parity-check code $C$. I understand that from the given parameters we can find its parity check ...
am567's user avatar
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is orthogonal complement of a subspace contained in another decomposition of Hilbert space

Let $H$ be a (infinite dimensional) Hilbert space and $v\in H$ be a nonzero vector. Define $V$ to be the span of $v$. It is given that $V+A=H$ where $A$ is a closed subspace of $H$. I am trying to ...
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Discrete Fourier Transform: choice of basis [closed]

I have two sets of N real numbers $\{E_m\}_m$ and $\{t_j\}_j$. I impose the following conditions: $\frac{1}{N} \sum_{m=1}^N e^{-i\,E_m(t_j-t_k)}=\delta_{jk} \hspace{1cm} \forall j, k$. $\frac{1}{N} \...
BlockSlicer's user avatar
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Uniquness of the orthogonality measure for generalized Laguerre polynomials [closed]

Let $\alpha>-1$. It is well-known that the measure $d\mu:=x^{\alpha}e^{-x}$ is the unique positive measure on $\mathbb{R}$ making generalized Laguerre polynomials $(L_n^{\alpha}(x))_n$ into an ...
user536450's user avatar
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Orthonormal basis for $\mathbb{C}^2$ over $\mathbb{R}$ [closed]

$\mathbb{C}^2$ is a 4-dimensional vector space over $\mathbb{R}$ with basis $\left\{\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} i \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}, ...
Mark Ren's user avatar
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Upper and lower bounding singular values of a nearly orthogonal matrix

Let $u_1, \dots, u_n$ be $n$-dimensional unit vectors and let $U = \begin{bmatrix}u_1 & \dots & u_n \end{bmatrix}$ be a matrix formed by stacking these vectors columnwise. If $u_i^\top u_j = 0$...
digbyterrell's user avatar
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Orthogonality of Whittaker functions

Is there a known orthogonality property of Whittaker functions $W_{\kappa,\mu}(iz)$ with respect to the first index as an integral over the argument? I am particularly interested in the case $\mu=0$ ...
Matt Majic's user avatar
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Deriving quadrature weights from discrete orthogonality of exponentials

In the proof of Lemma 2 of Driscoll and Healy, it says \begin{align} \sqrt{2}\delta_{k,0} &= \frac{1}{\sqrt{2}}\int_0^\pi P_k(\cos\theta)\sin\theta d\theta\\\\ &= \frac{1}{2\sqrt{2}}\int_{-\pi}...
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Finding the shortest distance from a point to a line

I've been using the following as a really good guide for this: Orthogonal projection of a point onto a line but I want to make sure that I have set this up, and understood it correctly. I have the ...
Calum's user avatar
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Complex $3\times 3$ matrix $A$ such that $A^TA=0$? [closed]

Can we find a non trivial complex 3x3 matrix $A$ such that $A^TA=0$? If I decompose $A$ as $B +iC$, with $B$ and $C$ real, I get 18 parameters with 18 equations corresponding to $B^TB=C^TC$ and $B^TC=-...
Laurent Jacques's user avatar
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Decomposition of primitive central idempotents in group algebras

Let $W$ be an irreducible $\mathbb{C}$-representation of a finite group $G$ with character $\chi_W$. A primitive central idempotents of the group algebra $CG$ is: $$e=\frac{\dim_{\mathbb{C}}(W)}{|G|}\...
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Characteristic polynomial of an orthogonal projection

Q: What is the characteristic polynomial of an orthogonal projection onto a (two-dimensional) plane through the origin in $\mathbb R^4$? Ans: $x^2(x-1)^2$ Can someone please explain how to do this ...
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Nature of the Euclidean Norm

I've been re-reading my linear algebra book and a definition is given of the norm of a vector in $\mathbb{R}^n$ to be: ||v|| $= \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}$. For $\mathbb{R}^2$ and $\mathbb{...
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The angle between u and v looks smaller than 90 degree but the dot product is still negative. [closed]

Screenshot of u and v graph Help
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Further decomposition of isotypic components in a representation

Let $(V,\rho)$ be an orthogonal (resp. unitary) representation of finite group $G$ whose irreducible representations over the same field as $V$ are $W_i$ with character $\chi_i$. We have $V \cong \...
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Trying to understand orthogonality of boundary conditions for functionals of the form $\int_{p_0}^{p_1}f(x,y)\sqrt{1+y'^2}dx$ bounded between 2 curves

A question I had whilst reading section 15 of Fomin's "Calculus of Variations" (great book btw!!) The General Question: Among all smooth curves whose end points $p_0$,$p_1$ lie between two ...
PhysicsIsHard's user avatar
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Meaning of a particular Mathematical symbol ($^\bot$) in Linear Algebra [duplicate]

While reading about Unitary Transformation between two sets of basis vectors of a vector space, I encountered the symbol $V^\bot$. Where, V is a Vector Space. What is its meaning? Thanks in advance.
Kalyan 's user avatar
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1 answer
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Substantiate the following statement " Arnoldi iteration is nothing but orthogonal projection onto Krylov subspaces"

Substantiate the following statement " Arnoldi iteration is nothing but orthogonal projection onto Krylov subspaces" My attempt Let $K_n=[\vec{b} | A\vec{b}| ... |A^{n-1}\vec{b}]$ be a ...
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Finding inner product associated with an orthogonal basis

Let $S = \{1 + k_0, c_1x + k_1, c_2x^2 + k_2, c_3x(x^2 - 3) + k_3\}$ ($x(x^2 - 3)$ is so that its derivative is proportional to $(1-x)(1+x)$), $-1 \leq x \leq 1$. Is it possible to find an inner ...
LaguerreGroup's user avatar
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1 answer
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Weight function given polynomial basis

Consider the following polynomial inner product: $$ (p,q)=\int_{-1}^{1} p(x) g(x) w(x) dx $$ It is well known that the polynomials of an orthogonal basis have simple roots, regardless of the weight ...
Jose's user avatar
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Polynomial formula for orthogonal vector in odd dimensions [duplicate]

I have been thinking about this problem recently. In 2 dimensions there is an easy formula for a nonzero vector orthogonal to a given vector $(x, y)$, namely $(-y, x)$. By taking pairs of coordinates, ...
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orthogonality in the hyerbolic geometry

Consider the half-plane model$ ~\mathbb{H}^2$ hyperbolic plane $\mathcal{P}:=i ,~\mathcal{T}:=10+i \text{ and } \mathcal{g}:=\{z\in \mathbb{H}^2:Re(z)=0\}$ Show that the straight lines$~\mathcal{PT}$ ...
tom31415's user avatar
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1 answer
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Condition of orthogonality of two solutions to a 2nd order ODE with constant coefficients on a smooth chart

I try to solve Exercise $100$ on page 37 of this pdf Let $\mathbf{r}:(u, v) \mapsto \mathbf{r}(u, v)$ be a smooth chart. Show that the solutions to the differential equation $$ A \dot{u}^2+2 B \dot{u}...
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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 compute Sylvester form of a matrix representing a symmetric bilinear form?

Can somebody state a step-by-step algorithm to, given a symmetric n x n-matrix A, (congruently) diagonalize A such that the entries of the diagonal are 1, -1 and 0 corresponding to the signature of A ...
romanson's user avatar
1 vote
1 answer
74 views

Orthogonal projection is bounded

Definition: Let $U$ be a subspace of $V$. The orthogonal projection of $V$ onto $U$ is the operator $P_U\in L(V)$ given by $$P_U(u+w)=u$$ if $u\in U, w\in U^{\perp}$. Let $V$ be a space with inner ...
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Why does $\iint_{\Bbb{R}^2}\ e^{-Q(x,y)}\ dxdy=\iint_{\Bbb{R}^2}d^2\hat{x}\ e^{-\lambda_1 \hat{\mathbf{x}_1}-\lambda_1 \hat{\mathbf{x}_2}}$?

Mi teacher wrote the following: $$\iint_{\Bbb{R}^2}\ e^{-(5x^2-4xy+5y^2)}\ dxdy=\iint_{\Bbb{R}^2}d^2\hat{x}\ e^{-\lambda_1 \hat{\mathbf{x}_1}-\lambda_1 \hat{\mathbf{x}_2}}=\frac{\pi}{\sqrt{\lambda_1 \...
MSU's user avatar
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1 answer
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How to proof that the perpendicular vector is perpendicular to all the vectors in the span and is not zero?

Let $v_1,\dots,v_p\in\mathbb{R}^m\setminus\{0\}$ be orthogonal and normalized. Let $V=\langle v_1,\dots,v_p\rangle$ be the span of $v_1,\dots,v_p$. Let $w\in\mathbb{R}^m\setminus V$ be another ...
David Krell's user avatar
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Linear Algebra: Orthogonal basis to find proj of $\mathbf{w}$ onto $\mathbf{y}$?

I'm currently studying for my final exam for linear algebra, and I'm a bit confused about how to find the projection of $\mathbf{w}$ onto $\mathbf{y}$. I already found the orthogonal basis for $W$, ...
ejry's user avatar
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3 votes
1 answer
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Transitivity of rational orthogonal matrices

It is well-known that the orthogonal group $O(n)$ acts transitively on the unit sphere in $\mathbb R^n$. What happens if we instead restrict ourselves to orthogonal matrices with rational coefficients?...
ViHdzP's user avatar
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2 votes
2 answers
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Equal angles are congruent, in an arbitrary symmetric bilinear space

Let $V$ be a finite-dimensional vector space with a symmetric bilinear form $B : V \times V \to F$. Consider two tuples $S = (u_1, u_2, \ldots, u_n)$ and $T = (v_1, v_2, \ldots, v_n)$, such that $$B(...
ViHdzP's user avatar
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1 answer
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Orthogonality of solutions to an eigenproblem

Let $\Omega$ be a connected, closed region in $R^2$, with $\Gamma$ being its boundary. ($\Gamma$ is piecewise smooth and non-self-intersecting but may not be necessarily connected - there may be one ...
G_B's user avatar
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1 vote
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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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0 answers
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How do you define a triangle from its orthographic projections on the 3 axis

I've spent hours on this problem and i'm running out of idea. Say you have an angle in a 3D space made of the points BAC, where A = (0,0,0), B and C can be moved and lengths AB and AC are always equal ...
Usylom's user avatar
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0 answers
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Find the matrix with respect to the canonical basis of the orthogonal projection of $R^3$ on $S$

I have a doubt. First I need to obtain the orthogonal projection of $R^3$ onto $S$, where $S$ is: $S=span{(1 0 -1),(0 1 -1)}$ and then the coordinate matrix with respect to the canonical basis of $R^3$...
ssj's user avatar
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5 votes
0 answers
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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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0 answers
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k-th Derivative of Legendre Polynomial is an orthogonal set

I have to show that the k-th derivative of the Legendre Polynomial $P_{i+k} , i \in \mathbb{N_0}$ is a set of orthogonal Polynomials with the weight function $\rho _k :(-1,1) \rightarrow \mathbb{R_+}:...
J3ck_Budl7y's user avatar
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Explanation of Linear Independence proof

Assume $v_1, .., v_n$ are nonzero and pairwise orthogonal, and suppose that $\alpha_1v_1 + \ .. \ + \ \alpha_nv_n = 0$ Then for each $j = 1, \ .., \ n$ $$ \begin{align} 0 & = \langle 0, v_j \...
InvestingScientist's user avatar
2 votes
3 answers
104 views

Why is $ R(A^*) \perp N(A)$ true?

Let a matrix the $A \in M_{n\times n}(\mathbb{C})$. My question is: (1) Why every matrix $A$ satisfies $ R(A^*) \perp N(A)$(where $R(A),N(A)$ are range of $A$,null space of $A$ respectively)? And why ...
user avatar
1 vote
1 answer
65 views

Finding Annihilation Operator in Space of Hermite Polynomials

This inquiry is related to my previously asked question entitled 'Proof of the Orthogonality of Hermite Polynomials' upon which I have defined a certain space with the following basis, $$ \psi_n(x) = [...
Hooman Puyandeh's user avatar
1 vote
0 answers
75 views

Proof of the Orthogonality of Hermite Polynomials

My question is regarding the proof of the orthogonality of Hermite polynomials. Actually, it's not quite the Hermite polynomials: $$ \psi_n(x) = [\dfrac{1}{\sqrt{n} 2^n n!}]^{\frac{1}{2}} e^{-\frac{x^...
Hooman Puyandeh's user avatar
1 vote
1 answer
91 views

How orthogonal projection connects with eigen space?

I asked this question and asked to @JonathanZ how orthogonal projection relates with eigen space, he gives me following replies in comments: Any time you have a subspace you can find an operator/...
user avatar
1 vote
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
0 votes
1 answer
49 views

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 ...
user avatar
0 votes
1 answer
42 views

An elementary inequality problem with 6 variables

I am trying to prove/disprove the following statement. Given $x,y,z \in \mathbb{R}$ and $0\leq a,b,c <1$ such that $x+y+z =0$ and $a^2x+ b^2y+c^2z<0$. Then $ax+by+cz<0$. In my context, $[x,...
abcxyzf's user avatar
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1 answer
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Prove P²=P for orthogonal projection formula

Given the following orthogonal projection formula $\hat{x}_w$ of $x \in \mathbb{R}^n$ on the vector $\frac{w}{\|w\|} \in \mathbb{R}^n$ defined by $\hat{x}_w := \frac{c \cdot w}{\|w\|}$ where $c = \...
fearloathing121's user avatar
1 vote
2 answers
94 views

Gram schmidt swapping two vectors

The question has background here but it's really just a linear algebra question. Suppose I have $B = (b_1,\cdots,b_n)$ vectors and I perform Gram Schmidt process (with no normalization of vector) ...
jacopoburelli's user avatar
0 votes
1 answer
59 views

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 ...
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