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where we used that the determinant of a [[diagonal matrix]] is the product of its diagonal elements and the fact that the determinant

...um <math>\scriptstyle \mathbf{R}_1^0,\ldots, \mathbf{R}_N^0</math>. The [[diagonal matrix]] '''M''' contains the masses on the diagonal. The diagonal matrix <math>\scriptstyle \boldsymbol{\Phi}</math> contains the [[eigenvalue]]s, w

...simply as scalar factor for each ''x''(similar to the multipliction with a diagonal matrix). Since ''w'' is real these factors remain unchanged in the adjoint operato ...l matrix does not commute with an arbitrary square matrix (except if the diagonal matrix is <i>c</i> '''E'''). This is the content of Schur's lemma.

...nd the [[trace (mathematics)|trace]] (Tr) of an operator is the sum of its diagonal matrix elements.