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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 saying that any such hyperplane can be written as the orthogonal space of a particular map. Then, I'm stuck: I've tried the case where this matrix is symmetric (and therefore can be written as a diagonal matrix over an orthonormal basis) in order to imitate the proof where you have to find an invertible matrix and not an orthogonal one.

I'm not sure where to go from here...

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  • $\begingroup$ If anyone speaks French, I've found this proof but it's very cumbersome, and is way too strong for what I'm trying to do... $\endgroup$ – John Do Feb 5 '18 at 16:48
  • $\begingroup$ I don't understand what you mean by a hyperplane contains a matrix. $\endgroup$ – thedude Feb 5 '18 at 16:50
  • $\begingroup$ @JohnDo What is proved there is that every hyperplane in $M_n(\mathbb{R})$ contains an orthogonal matrix. That makes sense. Your question doesn't. $\endgroup$ – José Carlos Santos Feb 5 '18 at 16:53
  • $\begingroup$ @thedude A hyperplane of the space of linear maps... It's a hyperplane over the space of all matrices $\mathcal{M}_n(\mathbb{R})$ (by isomorphism). $\endgroup$ – John Do Feb 5 '18 at 16:53
  • $\begingroup$ @JoséCarlosSantos Why does my question make no sense? The space of linear maps over $E$ contains orthogonal maps. One definition of an orthogonal map is that its matrix is orthogonal in every orthonormal basis. This problem can be looked at as matrices or as linear maps. $\endgroup$ – John Do Feb 5 '18 at 16:55
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The hyperplane can be written as the orthogonal of a particular matrix for the scalar product $<A,B> \mapsto Tr(^tA B)$.

If the matrix is symmetric, then it can be written as $A=^tP\Delta P$ where $P$ is orthogonal and $\Delta$ is diagonal. Let $M$ be a matrix containing four blocks, from left to right and top to bottom: zeroes, $I_{n-1}$, 1, zeroes.

It is indeed orthogonal and the $<\Delta,M>=0$ therefore $^tPMP$ is an orthogonal matrix in the given hyperplane.

In the general case, any matrix can be written as the product of an orthogonal matrix and a symmetric matrix $OS$. If $M$ is in the hyperplane orthogonal to $S$, then $^tOM$ is indeed symmetric and contained in the hyperplane orthogonal to $OS$.

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