# 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 ...
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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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### 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 $u_i^\top u_j = 0$...
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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 $\mu=0$ ...
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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 ...
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