# Antisymmetric Matrix Eigenvector Normalization

So, I have a complex $$4n \times 4n$$ antisymmetric matrix, $$A$$ and it has a non-degenerate spectrum. The matrix $$A$$ then has eigenvalues given by

$$\beta_{1}, -\beta_{1}, \beta_{2}, -\beta_{2}, ... , \beta_{2n}, -\beta_{2n}$$

where $$\text{Re}(\beta_{j})\geq0$$. $$A$$ has $$4n$$ eigenvectors $$v_{1}, ... , v_{4n}$$ with $$A v_{2j-1} = \beta_{j} v_{2j-1}$$ and $$A v_{2j} = -\beta_{j} v_{2j}$$.

Let $$V$$ be the matrix whose rows are the eigenvectors of $$A$$. Then it is possible to write $$A$$ as,

$$A = V^{T} \Lambda V$$

where $$\Lambda = \text{diag}(\beta_{1},...,\beta_{2n}) \otimes \begin{pmatrix}0&1\\-1&0\end{pmatrix}$$

In addition $$V$$ satisfies the property that $$VV^{T} = I_{2n} \otimes \begin{pmatrix} 0&1\\1&0\end{pmatrix}$$ ($$I_{2n}$$ is the $$2n \times 2n$$ identity matrix).

So Mathematica does not arrange the eigenvalues in the aforementioned pattern when I use Eigensystem and so $$V$$ does not satisfy the above property.

My question is: Is there an elegant way to order the eigenvalues in the way I mentioned so that $$V$$ has the above property?

Edit: Here is an example

n=5;x=RandomReal[1.,{2n,2n}];y=RandomReal[1.,{2n,2n}];

zero=Table[0,{i,1,2n},{j,1,2n}];

m = ArrayFlatten[{{-Transpose[x],y},{zero,x}}];
u = 1/Sqrt[2.]*KroneckerProduct[{{1,-I},{1,I}},IdentityMatrix[2n]];
A = ConjugateTranspose[u].m.u;
A = 1/2*(A-Transpose[A]);

{val,vec} = Eigensystem[A];

val//Chop

{5.07121, -5.07121, 0.0978161 + 0.839127 I, -0.0978161 - 0.839127 I,
0.0978161 - 0.839127 I, -0.0978161 + 0.839127 I,
0.103329 + 0.682027 I,
0.103329 - 0.682027 I, -0.103329 + 0.682027 I, -0.103329 -
0.682027 I, -0.495647 - 0.195394 I, -0.495647 + 0.195394 I,
0.495647 + 0.195394 I,
0.495647 -
0.195394 I, 0.513024, -0.513024, -0.325981, 0.325981, -0.251215,
0.251215}


As can be seen, the eigenvalues are not in the aformentioned order.

• One could use Ordering[] to sort the eigenvalues and eigenvectors in a desired manner. Do you have an example matrix? – J. M. will be back soon Sep 30 '18 at 2:36
• I can give the matrix I am dealing with but the code is a little lengthy (~ 40 lines). I can tell you that the matrix is complex antisymmetric as I said and its eigenvalues come in pairs of 4, $\beta_{j}, -\beta_{j}, \beta_{j}^{*}, -\beta_{j}^{*}$ (again $\text{Re}(\beta_{j}) \geq 0$). I'm having a bit of trouble giving an example matrix that satisfies this property that is succinct to post. – user1058860 Sep 30 '18 at 7:56
• I can speak only for myself, but without a concrete matrix I am so behind your state of information that I find it rather hopeless to successively mull over your problem... – Henrik Schumacher Sep 30 '18 at 10:04
• An succinct example matrix has been posted @J.M.issomewhatokay – user1058860 Sep 30 '18 at 21:41
• In your definition of $\Lambda$, the dimensions on the right-hand side don't match the left-hand side. Maybe you want a symplectic unit matrix there, and drop the negative eigenvalues? – Jens Sep 30 '18 at 22:30