Editing Skew-symmetric matrix (section)

=== Spectral theory ===

Since a matrix is [[matrix similarity|similar]] to its own transpose, they must have the same eigenvalues. It follows that the [[eigenvalue]]s of a skew-symmetric matrix always come in pairs ±λ (except in the odd-dimensional case where there is an additional unpaired 0 eigenvalue). From the [[spectral theorem]], for a real skew-symmetric matrix the nonzero eigenvalues are all pure [[imaginary number|imaginary]] and thus are of the form <math>\lambda_1 i, -\lambda_1 i, \lambda_2 i, -\lambda_2 i, \ldots</math> where each of the <math>\lambda_k</math> are real.

Real skew-symmetric matrices are [[normal matrix|normal matrices]] (they commute with their [[adjoint matrix|adjoints]]) and are thus subject to the [[spectral theorem]], which states that any real skew-symmetric matrix can be diagonalized by a [[unitary matrix]]. Since the eigenvalues of a real skew-symmetric matrix are imaginary, it is not possible to diagonalize one by a real matrix. However, it is possible to bring every skew-symmetric matrix to a [[block matrix|block diagonal]] form by a [[special orthogonal matrix|special orthogonal transformation]].<ref>{{cite book |first1=S. |last1=Duplij |first2=A. |last2=Nikitin |first3=A. |last3=Galkin |first4=A. |last4=Sergyeyev |first5=O.F. |last5=Dayi |first6=R. |last6=Mohapatra |first7=L. |last7=Lipatov |first8=G. |last8=Dunne |first9=J. |last9=Feinberg |first10=H. |last10=Aoyama |first11=T. |last11=Voronov |chapter=Pfaffian |chapter-url=https://link.springer.com/referenceworkentry/10.1007/1-4020-4522-0_393 |doi=10.1007/1-4020-4522-0_393 |editor-last=Duplij |editor-first=S. |editor2-last=Siegel |editor2-first=W. |editor3-last=Bagger |editor3-first=J. |title=Concise Encyclopedia of Supersymmetry |publisher=Springer |date=2004 |pages=298 |isbn=978-1-4020-1338-6 }}</ref><ref>{{cite journal|doi=10.1063/1.1724294|first=Bruno|last=Zumino|title=Normal Forms of Complex Matrices|journal= Journal of Mathematical Physics |volume=3|number=5|pages=1055–7 |year=1962|bibcode=1962JMP.....3.1055Z}}</ref> Specifically, every <math>2n \times 2n</math> real skew-symmetric matrix can be written in the form <math>A = Q\Sigma Q^\textsf{T}</math> where <math>Q</math> is orthogonal and
<math display="block">\Sigma = \begin{bmatrix}
  \begin{matrix}0 & \lambda_1 \\ -\lambda_1 & 0\end{matrix} &  0 & \cdots & 0 \\
  0 & \begin{matrix}0 & \lambda_2 \\ -\lambda_2 & 0\end{matrix} & & 0 \\
  \vdots & & \ddots & \vdots \\
  0 & 0 & \cdots & \begin{matrix}0 & \lambda_r\\ -\lambda_r & 0\end{matrix} \\
    & & & & \begin{matrix}0 \\ & \ddots \\ & & 0 \end{matrix}
\end{bmatrix}</math>

for real positive-definite <math>\lambda_k</math>. The nonzero eigenvalues of this matrix are ±λ<sub>''k''</sub> ''i''. In the odd-dimensional case Σ always has at least one row and column of zeros.

More generally, every complex skew-symmetric matrix can be written in the form <math>A = U \Sigma U^{\mathrm T}</math> where <math>U</math> is  unitary and <math>\Sigma</math> has the block-diagonal form given above with <math>\lambda_k</math> still real positive-definite. This is an example of the Youla decomposition of a complex square matrix.<ref>{{cite journal|doi=10.4153/CJM-1961-059-8|first=D. C. |last=Youla|title=A normal form for a matrix under the unitary congruence group|journal=Can. J. Math. |volume=13|pages=694–704 |year=1961|doi-access=free}}</ref>