Questions tagged [orthogonality]
This tag can be used to refer to the orthogonality of a set of vectors, a matrix or a linear transformation.
2,709
questions
0
votes
0
answers
8
views
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 $c Q_2$, ...
- 1
2
votes
1
answer
32
views
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 ...
- 329
0
votes
0
answers
14
views
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 ...
- 1,249
-1
votes
0
answers
35
views
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} \...
- 15
-1
votes
0
answers
40
views
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 ...
- 107
-1
votes
0
answers
39
views
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}, ...
- 1
2
votes
1
answer
30
views
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 $u_i^\top u_j = 0$...
- 300
2
votes
0
answers
24
views
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 $\mu=0$ ...
- 433
0
votes
0
answers
24
views
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}...
1
vote
1
answer
104
views
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 ...
- 399
0
votes
2
answers
82
views
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=-...
0
votes
0
answers
24
views
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|}\...
- 2,307
0
votes
1
answer
26
views
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 ...
- 637
3
votes
2
answers
114
views
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{...
- 197
-2
votes
1
answer
41
views
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
- 3