Questions tagged [orthogonality]
This tag can be used to refer to the orthogonality of a set of vectors, a matrix or a linear transformation.
0
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2answers
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Finding orthogonal basis in $\mathbb R^4$ from given vectors
I have two subsets of $\mathbb R^4$
$S=((-1,0,1,1),(0,1,1,1),(1,0,0,1))$
and
$T=(x,y,z,x-y+2z)$
I've proved that T is a subspace of $\mathbb R^4$ and that S is a basis for T. So far, so good!
I ...
-1
votes
2answers
34 views
The Order of Orthogonality [on hold]
I would like to show that $B\subset A$ implies $A^{\bot}\subset B^{\bot}$.
Note the meaning behind this: The bigger a subset, the smaller its orthogonal should be.
Let $x$ be in the complement of A. ...
2
votes
1answer
43 views
Finding orthonormal basis. Is there error on textbook?
The problem is finding orthonormal basis for W=span{u1=x,u2=x^2}
And as lots of people think, it is not very difficult problem
My answer is
${ \sqrt{3}x,\sqrt{80}(x^2-\frac{3}{4}x) }$
But, ...
3
votes
1answer
41 views
$H$ Hilbert space, $T$ symmetric bounded linear, when is $H=R(T) \oplus N(T)$?
I just saw in an exercise that if I have a prehilbert space $H$ and $T$ a linear, bound and symmetric operator then $R(T)=N(T)^{\perp}$. Now I was asking myself whether $H=R(T) \oplus N(T)$.
On wiki I ...
0
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0answers
13 views
orthogonal complement of subspace
I am not a mathematician but I faced the following problem when I want to apply some machine learning algorithm to solve nlp problem.
let say that I have a matrix $A$ that is $100X50$ and I project ...
0
votes
1answer
41 views
Find the equation of the plane that passes through the points
Find the equation of the plane that passes through the points:
$$P_1=(1,1,2)\\P_2=(2,3,3) \\P_3=(3,-3,3)$$
The answer writes:
Let $x=\vec{P_1P_2}=\begin{bmatrix}1\\2\\1\end{bmatrix}$ and $y=\vec{...
1
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1answer
20 views
Find orthogonal projection of $ [n,0,0,…,0]^T$ on subspace $V$
$n>1$ Given is $$V = \left\{ \vec{x} \in \mathbb R^n : x_1+x_2 + ... + x_n = 0 \right\} $$
a) Find orthogonal basis of $V^{\perp} $
b) Find orthogonal projection $\vec{x} = [n,0,0,...,0]^T$ on ...
1
vote
2answers
40 views
Find the orthogonal bases of the space $ V $ and $ V^{\perp}$
In space $\mathbb R^3 $ f Find the orthogonal bases of the space $ V $ and $ V^{\perp}$ where $$V = \left\{ \vec{x} \in \mathbb R^3 : x_1 - 3x_2 + x_3 = 0 \right\} $$
On the begining, I know that may ...
1
vote
1answer
36 views
Orthogonal complement of certain family of functions.
We know that the linear span of subspace $A$ in Hilbert space $X$ is dense if orthogonal complement of $A$, that is $A^{\bot}$ is trivial. I am given a family of functions $A=\{f_n\}$ and I was trying ...
0
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0answers
49 views
Prove that there exists a semi-orthogonal $U$ such that $U^TAU=B$, where $A$ and $B$ are positive-definite symmetric matrices.
Let there be a semi-orthogonal matrix $U \in \mathbb{R}^{m\times n}$ such that $U^TU=I_n$ if $m > n$
If $A \in \mathbb{R}^{m\times m}$ and $B \in \mathbb{R}^{n\times n}$ are positive-definite ...
1
vote
0answers
55 views
Projection associated to the decomposition $H=M⊕N$
Let $H$ be a Hilbert space and let $M$ and $N$ be two closed subspaces in $H$ such that $H=M⊕N$.
I'm trying to find a formula giving $P_{M,N}$ (the projection onto $M$ with respect to $N$) in terms ...
1
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0answers
13 views
Total subsets in incomplete inner product spaces [duplicate]
I'm reading Induced Representations of locally compact groups by Kaniuth and Taylor and I don't understand how total subsets work (in particular in Lemma 2.24).
I know that a total subset of an inner ...
0
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0answers
19 views
The definition of orthogonal complement in the column space
Denote $A^\bot$ is the matrix satisfied $A'A^\bot=0$ with the highest rank. Proof that:
(1)$I-(A')^-A'$ is a $A^\bot$, here $A^-$ means pseudo inverse.
(2)$M(A^\bot)=M(A)^\bot$. $M(A)$ is the column ...
0
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0answers
31 views
How to determine functions orthogonal to $\sin(a (T-t))$?
How to determine functions $u \in L^2 \cap L^\infty[0,T]$, $T > 0$, that are orthogonal on $[0, T]$ to $\sin\left[\frac{\pi^2 n^2}{l^2}\alpha \left(T - t\right)\right]$? In other words, how to ...
0
votes
2answers
20 views
Basis for orthogonal complement possibly with linear combinations
In $\mathbb{R}^4$, consider the subspace $W = Span(u_1, u_2,u_3)$ with
$$u_1 = (-1,1,0,0)$$
$$u_2 = (-1,0,1,0)$$
$$u_3 = (-1,0,0,1)$$
$(a)$ Use the Gram-Schmidt Process to ...
2
votes
1answer
36 views
Inner product with orthogonal complement
Let $\mathbb{R^3}$ be equipped with the inner product $<,>$ defined by setting
$$<\mathbf{u},\mathbf{v}>=2u_1v_1+5u_2v_2+3u_3v_3$$
for any pair of vectors $\mathbf{u}=(u_1,u_2,u_3)$ ...
0
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4answers
34 views
Orthogonal complement and norm
Can someone point out what I am fundamentally doing wrong in this question?
Consider the vector space $\mathbb{R}^3$ with the standard inner product (dot product) and let $H=span\left\{(2,1,0),(0,1,...
1
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1answer
31 views
On the construction of orthogonal polynomials
In the following proof, argument goes on based on considering $C_n$ to be nonzero then it finishes the proof for $C_n=0$ :
Also if we set $C_n=0$ in Eq. (6.10) then must $m=0,1,2,...,n- 2$ in Eq. (6....
1
vote
1answer
13 views
Why is the dot product of two columns $j$ and $k$ in FFT equal to $1+w^{j-k} + \dots + w^{(n-1)(j-k)}$?
The matrix is of the following form:
$\begin{pmatrix}
1 & 1 & \cdots & 1 & 1 \\
1 & w & \cdots & w ^{n-2} & w^{n-1} \\
\vdots & \vdots & \ddots & \vdots &...
0
votes
1answer
35 views
Existence of orthogonal matrices with zero diagonal and non-zero off-diagonal values
Does there exist an orthogonal matrix whose diagonal values are all zero but whose off-diagonal values are all non-zero for any $\Bbb R^n$?
Furthermore, does this conclusion change if we are talking ...
1
vote
2answers
56 views
Why $\int_0^{2\pi} \sin^2 x \mathrm{d}x \neq \int_0^{2\pi}\sin x \sin nx \mathrm{d}x \to n=1$
By orthogonality, we know that
$$\int_0^{2\pi}\sin mx \sin nx \mathrm{d}x = \pi$$
iff $m=n$ and $0$ otherwise. Nevertheless, when I am calculating it as follows, I get $0$.
$$\int_0^{2\pi} \sin x \...
0
votes
1answer
27 views
Infinite linear combination
I don't know how to start the proof of the following statement:
Let $V$ a real vector space with inner product $\langle\cdot,\cdot\rangle$. Consider a subset $C=\{v_j\}_J\subset V$ such that $\...
2
votes
2answers
27 views
$v \subseteq H \implies V^\bot$ is a closed subspace of the Hilbert space $H$
Exercise :
Show that if $H$ is a Hilbert space and $V \subseteq H$, then $V^\bot$ is a closed subspace of $H$.
Attempt :
I thought of two possible approaches. One would be a classic one, getting ...
1
vote
0answers
15 views
Orthogonality of integer shifts and sum of fourier transforms
A function $\psi \in L_2(\mathbb{R})$ is orthogonal to all integer shifts of a function $\varphi \in L_2(\mathbb{R})$ if and if only
$$\sum_{k\in \mathbb{z}} \hat{\varphi}(\xi+k)\overline{\hat{\psi}(\...
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1answer
10 views
Orthogonal transformation of standard normal sample
I was reading this pdf https://ocw.mit.edu/courses/mathematics/18-443-statistics-for-applications-fall-2003/lecture-notes/lec15.pdf
Shouldn't
$Var(Y_i)=\sum_{k=1}^nv_{ki}^2$ (from how $Y_i$ is ...
0
votes
1answer
25 views
Linear Algebra: orthogonality in $\mathbb{R}^{3}$
$$v_{1} = (1, 1, 1)$$
$$v_{2} = (-1, 1, 0)$$
$$v_{3} = (-1, -1, 2)$$
The vectors $v1$, $v2$, and $v3$ form an orthogonal basis for $\mathbb{R}^{3}$.
If $w = (4, -2, 4)$, then what is the ...
0
votes
1answer
28 views
Are circles also squares in $(\mathbb{R}^2,||\cdot||_{\infty})$?
In $(\mathbb{R}^2,||\cdot||_{\infty})$ circles appear to be squares. But do squares exist in general normed space when we do not have an inner product, hence no natural notion of orthogonality and ...
1
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0answers
33 views
Show that these curve families are orthogonal: $f(xy) = C$ and $y^2 - x^2 = D$ [duplicate]
Let the function $f : \mathbb{R} \rightarrow \mathbb{R}$ be differentiable and its derivative is never zero.
Show that these curves are orthogonal:
$$f(xy) = C$$
$$ y^2 -x^2 = D$$
$C$ and $D$ are ...
2
votes
1answer
43 views
Intersection of n hyperplanes in $\mathbb{R}^n$
For all unit vector $\nu \in \mathbb{R}^n$ consider an affine hyperplane $A_{\nu}$ orthogonal to the direction $\nu $.
Now consider n linearly independent unit vectors $\nu_ 1 , \nu_2, \dots, \nu_n \...
0
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1answer
22 views
Let $S$ be the subspace of $\mathbb{R}^4$ spanned by $x_1=(1,0,-2,1)^T$ and $x_2=(0,1,3,-2)^T$. Find a basis for $S_\perp$
Let $S$ be the subspace of $\mathbb{R}^4$ spanned by $x_1=(1,0,-2,1)^T$ and $x_2=(0,1,3,-2)^T$. Find a basis for $S_\perp$.
For this kind of question, if the subspace is spanned by one vector, I know ...
0
votes
1answer
25 views
Let $S$ be the subspace of $\Bbb R^3$ spanned by the vector $x= (x_1,x_2,x_3)^T$ and $y= (y_1,y_2,y_3)^T$
(a)
Let $S$ be the subspace of $\Bbb R^3$ Spanned by the vector $x= (x_1,x_2,x_3)^T$ and $y= (y_1,y_2,y_3)^T$, let $A =\begin{bmatrix}x_1 &x_2 & x_3 \\
y_1 &y_2 &...
1
vote
1answer
55 views
Show that $\langle x,y \rangle = 0$. [duplicate]
Suppose that $\left (X, \langle \cdot,\cdot \rangle \right)$ be a complex inner product space. Let $x,y \in X$ be such that $\|\alpha x + \beta y \|^2 = \|\alpha x\|^2 + \|\beta y\|^2$ for all pairs $\...
0
votes
3answers
48 views
Can we say that $x \perp y\ ?$ [closed]
Suppose $\left (X, \left < . ,. \right > \right)$ be a Hilbert space over $\Bbb C.$ Let $x,y \in X$ be such that $\|x+y\|^2 = \|x\|^2 + \|y\|^2$ $($ where $\|.\|$ is the norm induced by the ...
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0answers
16 views
Complex Endomorphism $f$ with orthogonal Eigenspaces implies that $f$ is unitary
I already posted this question earlier today without much success. After 3 hours and only 10 views I decided to give it another shot :) Of course I'll delete the second post!
Let $V \ne 0$ be an ...
1
vote
2answers
39 views
Finding perpendicular unit vector in $\mathbb{R}^n$ to hyperplane
Suppose I have $n-1$ linearly independent vectors $(v_1, ..., v_{n-1})$ in $\mathbb{R}^n$ that together form a basis of a hyperplane. I'm looking to find a last vector $v$ that is normal to the ...
2
votes
1answer
40 views
Some subspaces are either closed or dense
a) Let $a_{n}\rightarrow 0$ and $M=\{x\in \ell^{2}:\sum a_{n}x_{n}=0\}$. Show
that the subspace $M$ of $\ell^{2}$ is closed or dense according as $%
\{a_{n}\}\in \ell^{2}$ or $\{a_{n}\}\notin \ell^{2}$...
0
votes
1answer
17 views
How to prove symmetric matrix is orthogonally diagonalizable?
I learned below theorem and there is a proof that orthogonally diagonalizable matrix is symmetric, but there is no proof that symmetric matrix are orthogonally diagonalizable.
Theorem 2. An $n\...
2
votes
1answer
43 views
Damped vibrations of a membrane stretched over a circular frame
I am given this following PDE with the initial and boundary conditions with $0 < r < 1$, $t > 0$, and $v_0$ being a constant:
$u,_t,_t + 2bu,_t = u,_r,_r + \frac{1}{r} u,_r$
$u(t,r=0) = 0, \...
0
votes
0answers
19 views
converting trig function to orthogonal coordinates
How can a trig function (e.g. $r=\cos\theta$) be converted to orthogonal coordinates.
I'm imagining an infinite number of vector pairs forming $90^\circ$ angles but that doesn't feel right.
2
votes
1answer
60 views
Do every two orthogonal matrices in $\text{SO}(n)$ lie in the same coset of $\text{SO}(2)$?
Let $A,B \in \text{SO}(n)$.
Does there exist a homomorphism of Lie groups $\phi:\text{SO}(2) \to \text{SO}(n)$, such that $A,B$ lie in the same coset of $\phi(\text{SO}(2))\le \text{SO}(n)$?
0
votes
1answer
16 views
Equality of orthogonal projection norms. [closed]
Let $F,G$ be two sub vector spaces of $\mathbb{R}^n$, and denote $P_F,P_G$ the orthogonal projection matrices on $F$ and $G$ respectively. Suppose that for all $v \in \mathbb{R}^n$ we have:
\begin{...
0
votes
0answers
36 views
For which integers $k$ there exists an $(k-1)$-$(k, k, 1)$ orthogonal array?
This is what I've found out so far:
Since the first $k-1$ columns must have all the possible $k^{k-1}$ strings of length $k-1$ in exactly one row and the rows can be reorganized without changing the "...
0
votes
1answer
20 views
How to show that the bipolar co-ordinates are othogonal
How to show that the bipolar co-ordinates are othogonal
where $x=\dfrac{\sin hv}{\cos hv-\cos u},y=\dfrac{\sin u}{\cos hv-\cos u},z=z$
where $u\in [0,2\pi]$ and $y,z\in (-\infty,\infty)$.
...
1
vote
3answers
74 views
Determining a vector orthogonal to $q_1=(1,1,1)$ and $q_3=(1,1,-2)$, why I'm wrong with my calculations?
Consider the vectors $q_1=(1,1,1)$ and $q_3=(1,1,-2)$. I need to find a third vector $q_2$ such that $\{q_1,q_2,q_3\}$ is a arthogonal basis for $\mathbb{R}^3$.
My problem is the following: I did ...
0
votes
1answer
31 views
Orthogonal complement not resulting ok
I have the subspaces:
$$S = \langle(2,1,-1), (-1,2,0)\rangle, \qquad T = \{ X + Y + 2Z =0; X - Y - Z = 0\}.$$
I got that $T = \langle(1, 3,-2)\rangle$.
All vectors are linearly independent, so $S +...
1
vote
0answers
13 views
$L^{-1}$ in $LU$ decomposition relationship with $Q$ in $QR$ decomposition
Playing around with matlab, I see that the last row of $L^{-1}$ where $L$ is the part of the $LU$ decomposition is colinear with the last column of $Q$, in the matrix's $QR$ decomposition. Is there a ...
0
votes
0answers
19 views
Orthogonal complement of $H_a =\left\{g \in V: g\left(t+\frac{1}{\sqrt{2}}\right)=g(t) \right\}$
If $\;\;V=\{ f:\mathbb{R}\rightarrow \mathbb{C} |\; f \text{ is continuous and has period }1\}$, $\;\; \langle f | g \rangle$ is defined as $ \displaystyle \langle f | g \rangle = \int_0^1 \overline{f(...
1
vote
1answer
68 views
Orthogonal complement of $H_a =\left\{g \in V: g\left(t+\frac{1}{2}\right)=g(t) \right\}$
If $\;\;V=\{ f:\mathbb{R}\rightarrow \mathbb{C} |\; f \text{ is continuous and has period }1\}$, $\;\; \langle f | g \rangle$ is defined as $ \langle f | g \rangle = \int_0^1 \overline{f(t)}g(t)dt$, $\...
0
votes
0answers
29 views
Consider the plane P in R-3 given by x-y-2z=0
I found the matrix A whose columns are a basis for P, A=[1,-1,-2] (vertical form).
Using that I was able to find the projection matrix:
P=$\frac{-1}{2} \left( \begin{array}{cc} 1 & -1 & 2 \\...
0
votes
0answers
23 views
Orthogonality of generalized Newton symbol
Consider the functions $P_{n}(x)={x \choose n}.$ My question is, if there exists a measure $\mu$ with support being a subset of $(0,\infty)$ such that the family $\{P_{n}\}$ is orthogonal in $L^{2}(\...
