can you please show the proof of this theorem? Definition(Minimal Polynomial): Minimal polynomial of square matrix A is the monicpolynomial p(1) of least degree such that (1) = 0.Proposition: Two important facts regarding the eigenvalues of a matrix, A E cnxn (A may havereal, complex, distinct or repeted igenvalues):det(A) = ||Aii=1Trace(A) = dii=1

Answered: Image /qna-images/answer/72225e96-9695-4a90-9125-2cc7eab385ef.jpg

uare ma momial (2) of least degree such that $(1) = 0. position: Two important facts regarding the eigenvalues of a matrix, A E Cnxn (A may have ,complex, distinct or repeted igenvalues): det(A) = I[Ai Trace(A) = > %3D i=1

uare ma momial (2) of least degree such that $(1) = 0. position: Two important facts regarding the eigenvalues of a matrix, A E Cnxn (A may have ,complex, distinct or repeted igenvalues): det(A) = I[Ai Trace(A) = > %3D i=1

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

can you please show the proof of this theorem?

Definition(Minimal Polynomial): Minimal polynomial of square matrix A is the monic
polynomial p(1) of least degree such that (1) = 0.
Proposition: Two important facts regarding the eigenvalues of a matrix, A E cnxn (A may have
real, complex, distinct or repeted igenvalues):
det(A) = ||Ai
i=1
Trace(A) = di
i=1

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