If you carefully track the definition of the minimal polynomial of a linear operator or matrix, you will notice that the definition depends on whether we take F=R or F C. For instance, if A E Mnxn(R), then the minimal polynomial of A is a monic polynomial m e P(R) of smallest degree such that m(A) is the zero matrix. If we consider A as a matrix in Mnxn(C), then we are now looking for a monic polynomial m' e P(C) of smallest degree such that m'(A) is the zero matrix. If we take a fixed matrix A € Mnxn(R), under what circumstances is its minimal polynomial m with respect to R the same as its minimal polynomial m' with respect to C?
If you carefully track the definition of the minimal polynomial of a linear operator or matrix, you will notice that the definition depends on whether we take F=R or F C. For instance, if A E Mnxn(R), then the minimal polynomial of A is a monic polynomial m e P(R) of smallest degree such that m(A) is the zero matrix. If we consider A as a matrix in Mnxn(C), then we are now looking for a monic polynomial m' e P(C) of smallest degree such that m'(A) is the zero matrix. If we take a fixed matrix A € Mnxn(R), under what circumstances is its minimal polynomial m with respect to R the same as its minimal polynomial m' with respect to C?
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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Transcribed Image Text:If you carefully track the definition of the minimal polynomial of a linear operator or matrix, you will
notice that the definition depends on whether we take F R or F C. For instance, if A€ Mnxn(R).
then the minimal polynomial of A is a monic polynomial me P(R) of smallest degree such that m(A)
is the zero matrix. If we consider A as a matrix in Mnxn(C), then we are now looking for a monic
polynomial m'E P(C) of smallest degree such that m'(A) is the zero matrix. If we take a fixed matrix
A € Mnxn(R), under what circumstances is its minimal polynomial m with respect to R the same as its
minimal polynomial m' with respect to C?
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