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Unitary matrices have eigenvalues of unit magnitude, Hermitian matrices have Real eigenvalues. I think Unitary matrices and Hermitian matrices are also subgroups of Normal matrices. Therefore I think matrices which are both Unitary and Hermitian have eigenvalues 1 and -1 and are a subgroup of Unitary matrices and of Hermitian matrices. Do these matrices / this subgroup have a name?

An example such matrix is:

\begin{pmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & -1 \end{pmatrix}

Related:

Matrices which are both unitary and Hermitian

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  • $\begingroup$ @user953376 Thank you for your answer. Do you have a reference for this, just because I don't find many occurrences of that term in a web search? $\endgroup$
    – user83455
    Aug 22, 2021 at 16:46
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    $\begingroup$ also a type of involution. Typically reflection implies determinant of -1 which need not be true here. Why is this post tagged "finite-groups" -- there's no reason to think this is finite. And unless you constrain yourself to commuting matrices, this isn't a group. Why not test this out for yourself on 2 real Householder matrices? $\endgroup$ Aug 22, 2021 at 18:50
  • $\begingroup$ @user8675309 thank you. I'm still working this out. What group axiom is not held please? $\endgroup$
    – user83455
    Aug 22, 2021 at 19:00
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    $\begingroup$ it is not closed under products $\endgroup$ Aug 22, 2021 at 19:01
  • $\begingroup$ @user8675309 thanks that is very helpful, I did not realise. $\endgroup$
    – user83455
    Aug 22, 2021 at 19:01

2 Answers 2

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1.) $O_n(\mathbb R)$ is generated by Householder matrices.
2.) Every real Householder matrices is unitary and hermitian
3.) There are matrices in $O_n(\mathbb R)$ that have eigenvalues other than $\pm 1$ therefore the collection of matrices that are unitary and hermitian cannot form a group

These matrices are are all involutive. And if you e.g. constrain yourself to diagonal matrices, in the $2\times 2$ case you recover the Klein 4 group and for larger $n$ you recover what amounts to a generalization of Klein 4 group.

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I believe these are the reflection matrices. Specifically, generalised, orthogonal reflections.

They are generalised in the sense that they include reflections through the origin point, through a line through the origin, etc. not just a hyperplane of dimension one less than the containing space. ie. more than one axis can be reflected.

They are orthogonal in the sense that reflected directions are orthogonal to the 'mirror' hyperplane. This is the usual sense of the word reflection. It is distinguished from a possible more generalised type of reflection, where the direction of reflection through the mirror is at an angle to it.

This answer is really from the commenters, thanks to you!

As noted in the other answer, they are not a group because they are not closed under composition.

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