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linear algebra: diagonal matrices, orthogonal matrices, triangular matrices, symmetric matrices. The eigenvalues of those matrices are special too. I think 2 by 2 matrices provide terrific examples of the information that eigenvalues l can give. Sections 5.1 and 5.2 are worth careful reading, to see how Ax = lx is useful. Here is a case in which small

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Linear Algebra Igor Yanovsky, 2005 4 1 Basic Theory 1.1 Linear Maps Lemma. If A 2 Matmxn(F) and B 2 Matnxm(F), then tr(AB) = tr(BA): Proof. Note that the (i;i) entry in AB is Pn P j=1 fiijflji, while (j;j) entry in BA is m i=1 fljifiij. Thus tr(AB) = Xm i=1 Xn j=1 fiijflji; tr(BA) = Xn j=1 Xm i=1 fljifiij: 1.2 Linear Maps as Matrices Example.

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