Questions tagged [orthogonal-matrices]

Matrices with orthonormalized rows and columns. An orthogonal matrix is an invertible real matrix whose inverse is equal to its transpose. For complex matrices the analogous term is *unitary*, meaning the inverse is equal to its conjugate transpose.

453 questions
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How many degrees of freedom do orthogonal skew-symmetric matrices have?

$n$ by $n$ real orthogonal matrices have $n (n-1)/2$ degrees of freedom. So do the skew-symmetric matrices. But what about matrices that are both skew-symmetric and orthogonal? Is the number of such ...
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$3\times 3$ orthogonal matrix, which doesn't consist of zeros and ones [closed]

I'm stuck with my homework in a subject called Matrices in Statistics. Can you guys help with the following task? I would be very thankful! The task is as follows: Find a $3\times 3$ orthogonal ...
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Show non-singularity of orthogonal matrix

I'm given a question that says: A matrix $Q$ of size $n \times n$ is called orthogonal if its columns are orthogonal to each other and all columns have length $1$. a) Show that the matrix is non-...
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Every hyperplane contains an orthogonal matrix

Let $E$ be an euclidean space (over $\mathbb{R}$), I have to prove that every hyperplane of the linear maps over $E$ contains an orthogonal map (or equivalently, matrix). What I've tried doing is ...
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Which statement is false ?(Linear algebra problem)

Let $P=\dfrac{xx^{T}}{x^{T}x}$ be an a square matrix of order n where $x$ is a non zero column vector. Then which one of the following statement is False. $(A)$ P is idempotent $(B)$ P is ...
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A property of orthogonal matrices

Let $R$ be a $3\times 3$ orthogonal matrix. Let $v$ be the unit vector such that $Rv=v$ (upto sign change). Consider any unit vector $u$ such that $u^{T}v=0$ where $T$ stands for transpose. Show that ...
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Functions on $\mathbb{R}^n$ commuting with orthogonal transformations [duplicate]

I am interested in finding the functions $f:\mathbb{R}^n \to \mathbb{R}^n$ for which $f \circ U = U \circ f$ for all orthogonal transformations $U:\mathbb{R}^n \to \mathbb{R}^n$. Note that $f$ need ...
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Isotropic tensor field depending a vector

I am wondering how to prove the following statement (which is widely used, for example in turbulence theory) mathematically rigorously: Assume we are talking about $V=\mathbb{R}^3$. Given a tensor ...
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An orthogonal matrix which sends a vector to other vector with same length.

I know that For every to vectors $u,v\in \mathbb R^n$ where $|u|=|v|$ there exists an orthogonal matrix $A$ such that, $Au=v$. I have a problem so construct this matrix. is there any method to ...
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Deriving the Optimal Solution of the Orthogonal Procrustes Problem

I am trying to work through the Orthogonal Procrustes Problem but I do not understand a particular step. I would appreciate any help in understanding the steps the author goes from the first line to ...
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Show when the inequality for matrix-vector multiplication for the 2 norm is an equality?

I am asked to show for which vectors the inequality $||Ax|| \le ||x||||A||$ is an equality. My intuition tells me that this happens when $x$ is in the direction of the right singular vector ...
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Why eigenvalues of an orthogonal matrix made with QR decomposition include -1?

I want to make a real orthogonal matrix whose eigenvalues don't include -1. However, eigenvalues of a matrix $Q \in \mathbb{R}^{n\times n}$ ($n$ is even number) made with QR decomposition of a random ...
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Let $A(\theta)$ be a given function , where $\theta \in (0, 2\pi)$. Mark the correct statement below

Let $A(θ) = \left[ {\begin{array}{cc} \cosθ & \sinθ \\ -\sinθ & \cosθ \\ \end{array} } \right]$ where $θ ∈ (0, 2π)$. Mark the correct statement below A. $A(θ)$ has ...
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4 dimensional orthogonal group

Let's say I have a 4 by 4 matrix that is in the orthogonal group. The first three columns A, B, and C are known. Now, I can do a system of equations (4 equations) to solve for D, the fourth column. ...
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Orthogonal Group

(Note: $O(n) = O(n, \Bbb{R})$) I've notice that $O(1)$ is equivalent to $S^0$. And I've read that $O(2)$ is equivalent to two copies of the circle group $S^1$. I was wondering if someone can ...
Nearest Semi Orthonormal Matrix Using the Entry Wise ${\ell}_{1}$ Norm
Given an $m \times n$ matrix $M$ ($m \geq n$), the nearest semi-orthonormal matrix problem in $m \times n$ matrix $R$ is \begin{array}{ll} \text{minimize} & \| M - R \|_F\\ \text{subject to} &...
I'm trying to prove the following statement: if A and B are unitarily equivalent, then they have the same singular values so my proof goes like this: unitarily equivalent means $A=QBQ^*$ so if \$A=...