# 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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### What is the spectral family of $\left[ \begin{array}{cc} 0&1&0\\ 1&0&0\\ 0&0&1 \end{array} \right]$?

Spectral Family:A real spectral family (or real decomposition of unity) is a one parameter family $\mathcal E=(E_{\lambda})_{\lambda\in \mathbb R}$ of projections $E_{\lambda}$ defined on a hilbert ...
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### Action that involves a rotation on the real hyperplane

For an element $v \in V$ with $d(v,v) = 1$ (i.e. $v$ is a unitary vector), define $r_v \in \mathrm O(V)$ by $$r_v(w) := -\rho_v(w) := 2d(v,w)v - w.$$ Here $\rho_v$ is the reflection in the real ...
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### Extension of Wahba’s problem with infinite weights

Wahba's problem seeks to find a $3 \times 3$ orthonormal rotation matrix that minimizes $$J(\mathbf{R}) = \frac{1}{2} \sum_{k=1}^{N} a_k\| \mathbf{w}_k - \mathbf{R} \mathbf{v}_k \|^2$$ I want to ...
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### Prove existance of orthonormal basis of $\mathbb{R}^3$ consisting of eigenvectors of generalized eigenvalue equation

Context I am studying normal modes oscillations and normal modes [1,2]. In an earlier post , I asked for a proof that the generalized eigenvalue equation in normal-mode analysis has positive ...
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### Row (or column) sums of orthogonal matrix (excluding scalar multiples of identity matrix) always different

Are row (or column) sums of orthogonal matrix (excluding scalar multiples of identity matrix) always different? Suppose $Q$ is an $p\times p$ orthogonal matrix that is not a scalar multiple of ...
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### Is there are a $3 \times 3$ real, orthogonal matrix $Q$ that has exactly three zero entries? [closed]

If I have choose first column with two zeros, then other two columns have two or none. Or else there are are three zeros in each column. I don't know how to conclude after this
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### If an orthogonal matrix represents a reflection, show that it is symmetric. [closed]

The question is as it says. I am a first-year uni undergraduate student for context. I don't really know how to approach this question. Any hints/suggestions would be greatly appreciated!
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### Is the external direct product of O(2) with itself isomorphic to O(2)? (Orthogonal group of 2 x 2 M-matrices)

Is the external direct product of $O(2)$ with itself isomorphic to $O(2)$ i.e. $O(2) \oplus O(2) = O(2)$ I'm guessing they're not, as to prove that is was, I'd have to define an explicit isomorphism ...
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### Prove that [O(2), O(2)] = SO(2). The commutator subgroup of the ortogonal group is equal to the special orthogonal group.

Prove that $[O(2),O(2)] = SO(2)$. In words, prove that the commutator subgroup of the orthogonal group of 2x2 matrices is equal to the the Special Orthogonal Group of 2x2 matrices. I know that the ...
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### Calculate rotation matrix to rotate a matrix A (3d points x,y,z) to be orthogonal by 3D normal vector N [closed]

I have a matrix A, which is the coordinates of a circle in 3D space. I want to rotate the circle in a way that its normal vector (orthogonal to the circle) be aligned with vector N:(x_n,y_n,z_n). I'd ...
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What maps $h:\mathbb{R}^N\to\mathbb{R}^N$ exist such that for $A,B\subset\mathbb{R}^N$ we have $A\sim B \iff hA\sim hB$. There is a trivial solution when $h$ is a similitude: $h(x)=rOx+t$ where \$r\in \...