The goal of this problem is to show that any n n matrix A is similar to a triangular matrix (here we allow entires to be complex numbers). The proof is by induction on n.

a) Show the statement for n = 1

b) Show that there always exist an eigenvector u1 and corresponding eigenvalue 1 for A (complex entries are allowed).

Show that A is similar to a matrix of the form 1 w 0 B1 , where 0 is the zero vector with n 1 rows, w is some 1 (n 1) matrix, and B1 is some (n 1) (n 1) matrix.

c) By induction there exist an invertible matrix P1 such that B1 = P1T1P 1 1 , where T1 is a triangular matrix.

Show that A is similar to a triangular matrix. Hint: 1 w 0 C1C2C3 = 1 0 0 C1 1 w 0 C2 1 0 0 C3

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The goal of this problem is to show that any n x 71 matrix A is similar to a triangular matrix (here we allow entires to be complex numbers). The proof is by induction on n. a) Show the statement for n = 1 b) Show that there always exist an eigenvector 17:1 and corresponding eigenvalue A1 for A (complex A61 31] , where [l is the zero vector with n 1 rows, to is some 1 x (n 1) matrix, and Bl is some (n 1) x (n 1) matrix. entries are allowed). Show that A is similar to a matrix of the form [ c) By induction there exist an invertible matrix P1 such that Bl = P1T1P1 1, where T1 is a triangular matrix. Show that A is similar to a triangular matrix. Hint: A1 11) _10 Ala; 10 {l 010203001 0 02 003