Advanced Linear Algebra, Lecture 6.3: Normal linear maps

Advanced Linear Algebra, Lecture 6.3: Normal linear maps

Previously, we showed that commuting diagonalizable linear maps are simultaneously diagonalizable, which means that they share a common basis of eigenvectors. If these maps are self-adjoint, then they share a common orthonormal basis of eigenvectors, or in our language, a common spectral resolution. This is key to understanding precisely which linear maps have an orthonormal basis of eigenvectors, besides self-adjoint, anti-self-adjoint, orthogonal, and unitary maps. The answer are linear maps that are normal, which means they commute with their adjoints. This condition is key because if we decompose a linear map into its adjoint and self-adjoint part, as M = (M+M*)/2 + (M-M*)/2 = H + A, then M*M=MM* is precisely what is needed for H and iA to be commuting self-adjoint maps, and then we apply our previous result. We conclude this lecture by establishing several properties of normal and unitary linear maps.

[artcellCTRL]

sir, are you going to upload the rest of the lectures of that course in the future ? [thank you for the lectures that you have uploaded so far]

[SilverlightLantern]

17:03 Clarification: I think (iA) is self-adjoint also.