6. [Aug. 31.] To Use the Jordan Form or Not to Use Jordan Form. Sometimes the use of the Jordan...

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6. [Aug. 31.] To Use the Jordan Form or Not to Use Jordan Form. Sometimes the use of the Jordan Canonical Form and matrices with multiple eigenvalues can be avoided using the following considerations.
(a) Let A E Mnxn(C)• Show that given c > 0 there exists a matrix B with distinct eigenvalues so that IIA - Bil
P-1AP =U =
1111 U12 ... Uin
0 1122 . • • U2n
XXXXXXXXXXunn j
Show that this fact can be used instead of Jordan Form to characterize all solutions of Y = Ay (as linear combinations of products of certain exponentials, polynomials and trigonometric functions). [c.f., Bellman, Stability Theory of Differential Equations, pp. 21-25.] )
(d) Let A E •nxn(C)• Show that given E > 0 there exists a nonsingular P such that in addition to (1) we may arrange that Eif (AB) = f (A) f (B) for all A, B.
You can probably find several different arguments on your own. [ibid.; or Kurosh, Higher Algebra, p. 334.]

005_xwe89zk-nkg1oinz.docx

Robert · Accepted Answer

Remark: For matrix decomposition, there are a lot of ways to decompose
it. For example: Schur Decomposition, Jordan form, Diagonalization, LU de-
composition etc. Proof of existence of all kind decompositions assume existence
of one decomposition and then prove for other. For this assignment: I will
assume that there exists Schur Decomposition. In the problems, there is no
restriction for assumption. Hence I will solve this assignment problem assuming
Schur decomposition:
Schur Decomposition: After a suitable change, any matrix can be written
as
A = D +N
where D is diagonal matrix and N is strictly upper triangular matrix.
a. By Schur Decomposition, we have
A = D +N
If D = diag(d1, .., dn),

6. [Aug. 31.] To Use the Jordan Form or Not to Use Jordan Form. Sometimes the use of the Jordan Canonical Form and matrices with multiple eigenvalues can be avoided using the following considerations....

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