# 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.

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### Show that the matrix $P=I-2hh^T$ is orthogonal and find its first column.

Let $x=(x_1,...,x_n)^T$ a column vector in $\mathbb{R}^n$ so that $x_1\neq -1.$ Let $h$ a unitary vector in the direction of $x-e_1$ where $e_1$ is the vector in $\mathbb{R}^n$, $e_1=(1,0,...,0)$. ...
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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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### Circulant Orthogonal $\operatorname{MDS}$ Matrix

Definition: A matrix $M$ of order $n$ over a field is a $\operatorname{MDS}$ matrix if and only if every sub-matrix of $M$ is non-singular. My question: How to proof the following statement. If $A$ ...
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### Orthogonal columns of a nonsquare matrix

Consider a SVD of a $3 \times 2$ matrix $A$, why the product of a $2 \times 2$ $S$ orthogonal matrix $AS$ has orthogonal columns while the product of a $3 \times 3$ orthogonal matrix $SA$ won't? I ...
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### None element of orthogonal matrix can't have unit modulus larger then 1

None element of orthogonal matrix can't have unit modulus larger then 1. I've tried to use the properties of orthogonal matrices ( $|det(A)| = 1$ and $Q^T=Q^{-1}$ ) but I couldn't find out how they ...
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### Uniformly random matrices in SO(n)

I don't know whether this question fits best on StackOverflow, Math.SE or CrossValidated; I am looking for a practical algorithm to generate uniformly distributed matrices in $SO(n,\mathbb{R})$ (aka ...