By Ghislaine A.

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**Sample text**

By induction on n 2 N. The case n D 1 is trivial. Let now n 2 N and assume that any A 2 C n n is unitarily similar to an upper triangular matrix. nC1/ . Due to the fundamental theorem of algebra, we can ﬁnd 2 C with pA . / D 0. Since therefore I A is not injective, we can ﬁnd a vector q 2 N . I A/ n ¹0º with kqk D 1. Obviously, q is an eigenvector for 28 Chapter 2 Existence and properties of eigenvalues and eigenvectors the eigenvalue . nC1/ (cf. qN 1 / P q D ı1 : and B 2 C 1 n satisfying Â Ã B : P AP D b A n Since b A is only an n n matrix, we can apply the induction assumption and ﬁnd a b 2 C n n with b 2 C n n and an upper triangular matrix R unitary matrix Q b b b D R: b Q AQ Now we let Q WD P and observe Q AQ D D Â 1 !

47 (Order of eigenvalues). A/ into the upper left entry of R, and by extension we can choose any order for the eigenvalues on the diagonal of R. 48 (Trace). , by the sum of diagonal entries. Let Q 2 F n n be unitary. , that the trace is invariant under similarity P transformations. Q AQ/ D ni;j;kD1 qNj i qki aj k and considering the ﬁrst two factors. , for ﬁnding all eigenvalues and the corresponding algebraic multiplicities, we are mainly interested in diagonalizing a matrix. 49 (Normal triangular matrix).

D 0. Since therefore I A is not injective, we can ﬁnd a vector q 2 N . I A/ n ¹0º with kqk D 1. Obviously, q is an eigenvector for 28 Chapter 2 Existence and properties of eigenvalues and eigenvectors the eigenvalue . nC1/ (cf. qN 1 / P q D ı1 : and B 2 C 1 n satisfying Â Ã B : P AP D b A n Since b A is only an n n matrix, we can apply the induction assumption and ﬁnd a b 2 C n n with b 2 C n n and an upper triangular matrix R unitary matrix Q b b b D R: b Q AQ Now we let Q WD P and observe Q AQ D D Â 1 !

### La Pratique de la Correspondance by Ghislaine A.

by Robert

4.3