Characteristic and minimal polynomials

Dear Professor Kim,

Can you show me how to find the JNF of matrix A in the exam paper 2005 (Q6,part(a)-(v))

The matrix A is given by A=diag(Matrix B,Matrix C). i.e.:

A=diag( row1(1,2,2),row2(0,1,2),row3(0,0,2); row1(2,1,1),row2(0,2,1),row3(0,0,1) )

How can i find ch_A(x) and m_A(x) in this case?

Moreover,in general,i am wondering whether there is a quick way to find out the characteristic and minimal polynomials for a matrix? For example,if i was given a 3×3 matrix T that i cannot write down the characterstic straight away,so do i have to go through the whole process about how to calculate the det(xI-T) in order to find ch_T? or there is an easier way?

Also for the minimal polynomial,how could i know it is equal to the characteristic or not? if it is not ,then how can i obtain it easily?

I know if CH has no multiple roots,then M=CH. But,how about when CH has multiple roots. e.g.:CH=(x-1)^3(x-2)^3,in this case how can i compute the M?

Your help would be much appreciated!

Many Thanks!

Reply:

There’s no really easy way in general to calculate the characteristic polynomial other than computing the determinant. However, if you recall general properties, there might be easier or harder ways to calculate the determinant. For example, for your A above, it is clear that

ch_A(X)=ch_B(X)ch_C(X)

The minimal polynomial also needs to be calculated from the definition in general. If

ch_T(X)=(x-1)^3(x-2)^3

you know that the possibilities for the minimal polynomial are

(x-1)(x-2), (x-1)^2(x-2), (x-1)(x-2)^2, (x-1)^2(x-2)^2, etc.

since it divides ch_T and has the same roots. Which one it is must be determined in general by plugging in the matrix T for X, and finding the one of minimal degree for which the resulting matrix is 0.

If you think about the definition carefully, there are occasional shortcuts. For matrix A above,

m_A(X)=LCM(m_B(X),m_C(X))

where LCM refers to the least common multiple. What does this say if m_B and m_C have no common roots?

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