# 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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### Set invariant under group action

I am reading a paper with the following description: $O(n): \{Y\in \mathbf{R}^{n\times n}\mid Y^TY=I\}$ We say a set $V$ is $T$-invariant if $TV\subseteq V$, where $T$ is a linear transform. So ...
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### Improper rotation matrix in $2D$

The following is the related problem: Improper Rotations in Even Dimensions I want the simpler explanation. An improper rotation is rotation, followed by reflection in the plane perpendicular ...
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### Are the groups $G:=\{A \in M(n,\mathbb R) : A=A^t\}$ i.e. the group ( under addition ) of symmetric matrices and $O(n,\mathbb R)$ isomorphic?

Let $G:=\{A \in M(n,\mathbb R) : A=A^t\}$ i.e. the group ( under addition ) of symmetric matrices ; Are $G$ and $O(n,\mathbb R)$ isomorphic ?
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### Finding an orthogonal matrix for a 3x2

I know how to find an orthogonal matrix for a $2\times2$ or $3\times3$ matrix. However I have been stuck on how to do this for a $3\times2$ matrix. The question is how to find a non-zero $3\times2$ ...
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### What exactly does a rotation preserve?

I understand a rotation should preserve length and angle and hence the dot product. Since anything that preserves the dot product is a linear transformation, then a rotation can be represented by a ...
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### For an orthogonal matrix $Q$, why does $QQ^T = I$?

In my linear algebra text (Strang), an orthogonal matrix is defined to be a square matrix whose columns are orthonormal. In other words, an orthogonal matrix is a matrix $Q = [q_1 \cdots q_n]$ where ...
I'm trying to solve 3 problems on special orthogonal groups, and I need proof verification of the first 2 and help with the proof of the 3rd. Consider $SO(n)$ the set of all $n \times n$ matrices ...