tl;dr: How do I construct the symplectic matrix in Williamson’s theorem? I am interested in a constructive proof/version of Williamson’s theorem in symplectic linear algebra. Maybe I’m just missing a simple step, so here is what I know: Let us fix the symplectic form $J=\begin{pmatrix} 0_n & 1_n \\ -1_n & 0_n\end{pmatrix}$. Recall: Theorem: Let […]

Bilinear forms can give us a notion of distance, whether it is the typical Euclidean distance, or the spacetime interval between two events in Minkowski space. But what about skew-symmetric bilinear forms? Skew-symmetry means that every vector has $B(v,v)=0$. Also, we can always find a basis such that picking any one element of that basis, […]

So it’s been a while since I’ve taken Linear Algebra, but my friend asked me a question, that I couldn’t answer. If a matrix $A$ exists such that $A^3 = I$, does $A$ have to equal the identity matrix $I$? My first instinct was to say no, but… (edited out my incorrect math) EDIT: thanks […]

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