When dealing with $\Bbb{R}$ it is easy to visualise matrices as how they 'distort' space - I am wondering if, when dealing with $\Bbb{C}$ there is also a geometric way of visualising these matrices, or at least some sort of intuition behind them besides how the linear transformation acts on the basis. Specifically, what is the intuition/geometric visualisation behind a hermitian matrix and a positive definite matrix (e.g. an orthogonal matrix geometrically represents rotation/reflection/etc.). Cheers.

  • $\begingroup$ The fact that Hermitian matrices can be diagonalized into a real diagonal matrix by a unitary matrix gives some intuition. In particular, this says there's always an orthogonal basis with real eigenvalues ... So we know our space decomposes into a direct sum of orthogonal subspaces in such a way that the Hermitian operator scales each summand by some real number. $\endgroup$ – Lorenzo Aug 4 '18 at 23:23

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