Diagonalization and Inner productsThe Spectral Theorem for Normal matrices*Orthogonality of eigenvectors applies to a larger class of matrices, but it isonly the symmetric case that guarantees diagonalizability overR.Nevertheless, for real matrices that are diagonalizable overCbut notR,we had a nice structure theorem (thenormal form), which decomposedtheir action onRninto lines and planes. When is that decompositionorthogonal, and moreover, when do the plane actions involve genuinerotations and scalings, as opposed to ‘elliptic’ rotations?Theorem (Spectral theorem for normal matrices)The normal form A=SRS-1of a real matrix can be obtained with anorthogonal S iff A is anormalmatrix, that is, A commutes with AT.The proof, like the description of the real normal form, requires the use ofcomplex numbers and complex inner products; I will omit it.Normal matrices include the importantorthogonal matrices: ifQT=Q-1,thenQTQ=In=QQT. In particular, rigid motions of Euclidian space (in3 or more dims) take their normal form in an orthonormal basis.