Collected mathematical papers, volume 3 by Sylvester J.J.

By Sylvester J.J.

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This is the fundamental theorem of algebra. 7 of Chap. 3. 6, is given in one of the exercises at the end of this section. 28) dy B D@ dt 1 a11 :: C A y; : ann you can solve the last ODE for yn , as it is just dyn =dt D ann yn . 21), and you can continue inductively to solve. Thus, it is often useful to be able to put an n n matrix A in upper triangular form, with respect to a convenient choice of basis. We will establish two results along these lines. The first is due to Schur. 7. For any n n matrix A, there is an orthonormal basis u1 ; : : : ; un of C n with respect to which A is in upper triangular form.

39) B m D 0 for some m Ä k: 4. Constant-coefficient linear systems; exponentiation of matrices 23 Proof. C k /; then C k W1 W2 is a sequence of finitedimensional vector spaces, each invariant under B. This sequence must stabilize, so for some m; B W Wm ! Wm bijectively. If Wm ¤ 0; B has a nonzero eigenvalue. We next discuss the famous Jordan normal form of a complex n The result is the following. n matrix. 13. 12, it suffices to establish the Jordan normal form for a nilpotent matrix B. Given v0 2 C k , let m be the smallest integer such that B m v0 D 0I m Ä k.

R2n be open, 3. Identify R2n with C n via z D x C iy, as in Exercise 4 of 1. Let U and let F W U ! R2n be C 1 . p/ is invertible. If F 1 W V ! 1, show that F 1 is holomorphic provided F is. 4. Let O Rn be open. x ˛Š x0 /˛ ; ˛ 0 valid in a neighborhood of x0 . 14), and obtain an extension f to a neighborhood of O in C n . Show that the extended function is holomorphic, that is, satisfies the Cauchy–Riemann equations. Remark. It can be shown that, conversely, any holomorphic function has a power-series expansion.

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