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I need some help analyzing the variety of normal matrices $M M^\dagger -M^\dagger M = 0$. Each entry in this equation satisfies a quadratic equation $$\sum_j h_{ij} \overline{h_{kj}} = \sum_j \overline{h_{ij}} h_{kj} $$ Where $i,j = 1, \dots, n$. This reminds me of the Plucker embedding of the Grassmanian as the intersection of quadratics.

If this were a single non-degenerate conic it would be similar to some kind of cone $x_1^2 + \dots + x_k^2 - x_{k+1}^2 -\dots -x_n^2=0 $. But this is an intersection of (possibly degenerate) quadrics. Not at all sure the structure of this variety.

I'm looking for general information about this variety. Is this variety reducible or singular? What kind of components does it have? What's it's dimension?


EDIT: By the spectral theorem normal matrices are similar to diagonal matrices, $\mathrm{diag}(\lambda_1, \lambda_2, \dots, \lambda_n)$ by a unitary matrix so there is a $U(n)$ action on the variety of normal matrices fibered by the diagonal matrices themselves. Some of these orbits are degenerate (like when some $\lambda = 0$).

So maybe $\dim \{ [M, M^\dagger]=0\} = \dim U(1)^n + \dim U(n) = n^2 + n$

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You have to be careful with what you mean here. As your equations involve complex conjugation they do not define a complex variety. They do define a real algebraic variety. However, then you have to be careful because there may be reducible components that "are not seen" easily in the real picture, they may not have real points, only complex ones (to be precise all their real points may lie on othere irreducible components). This is what happens in your example.

The first step in understanding a real algebraic variety is usually to extend scalars to the complex numbers. To understand what happens in your case it pays to describe a little bit more abstractly. Hence considered over the reals what we have is a finite dimensional $\mathbb R$-algebra with an $\mathbb R$-involution, namely $\mathbb C$. We then consider $n\times n$-matrices over $\mathbb C$ which inherits an involution from the involution of $\mathbb C$, $M\mapsto M^\dagger$. As it is $\mathbb R$-linear the condition $MM^\dagger=M^\dagger M$ is given by polynomials with real coefficients and hence defines a real algebraic subvariety of $M_n(\mathbb C)$ which is seen as the affine space $\mathbb A^{2n^2}_{\mathbb R}$ over the reals. When we extend scalars we get exactly the same description but replacing the involutive $\mathbb R$-algebra $\mathbb C$ with its scalar extension $\mathbb C\bigotimes_{\mathbb R}\mathbb C$. This algebra is isomorphic to $\mathbb C\times\mathbb C$ with the involution permuting the two factors. Hence $M$ is now described by two complex matrices $(A,B^t)$ and $(A,B^t)^\dagger=(B,A^t)$ and $MM^\dagger=M^\dagger M$ is turned into $(AB,B^tA^t)=(BA,A^tB^t)$ which just describes pairs $(A,B)$ of commuting matrices. Hence the complex scalar extension is the variety of pairs of commuting matrices with a non-standard real structure where complex conjugation takes $(A,B)$ to $(\overline B,\overline A^t)$ (instead of $(A,B)\to (\overline A,\overline B)$ which is the more standard one).

Now, it is well-known that the variety of pairs of commuting matrices is very complicated. There is a more easily understood subset consisting of the pairs $(A,B)$ where both $A$ and $B$ are semi-simple. From an algebraic point of view it is not as it a subvariety (it is a constructible but not locally closed subset). It does contain the open algebraic subset of pairs $(A,B)$ where all the roots of the characteristic polynomials of $A$ (say) are distinct. That subset is Zariski dense in the set of all semi-simple matrices and hence the latter lie in the closure of the former. All the real points will then lie in this closure $\overline S$. However, it is well-known that unless $n$ is very small the closure is not equal to the whole variety (there are pairs that can not be deformed into semi-simple ones). This means that while $\overline S$ is an irreducible component (that is defined over the real numbers) there are (many) other irreducible components. They will be exactly of the type described above, all their real points will also lie in $\overline S$.

Addendum: As to the question of dimension, of course the closure of semi-simple matrices has the right dimension but I think (but don't quite remember) that there are other components of larger dimension.

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If you need a reference, most of what Torsten Ekedahl explains above can be found in the paper

Kh. Ikramov, The dimension of the variety of normal matrices, Zh. Vychisl. Mat. Mat. Fiz., 38 (1998), 5–10

In fact Ikramov has quite a few papers devoted to similar problems (for example for the variety of conjugate normal matrices which satisfy $XX^{\dagger}=\overline{X^{\dagger}X}$ etc.). The paper above is hard to find in electronic form, I believe, but some of its results are reproduced in S. Friedland's "Normal Matrices and the Completion Problem" (which is also available here). In particular Lemma 2.4 proves that the variety of (complex) normal matrices as a real variety is irreducible of dimension $n(n+1)$, and its set of smooth points corresponds to normal matrices with distinct eigenvalues.

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    $\begingroup$ Just to emphasise the reality of possible confusions, Ikramov uses irreducibility in the sense of the Zariski topology induced on the real points of a real algebra variety and not in the sense of irreducibility of the whole variety. This is quite common in real algebraic geometry. $\endgroup$ – Torsten Ekedahl Nov 8 '11 at 7:58

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