Show that minimal poly for a sq matrix and its transpose is the same

  • #1
catsarebad
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Homework Statement


show that minimal poly for a sq matrix and its transpose is the same

Homework Equations



The Attempt at a Solution


no clue.
 
Last edited:
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  • #2
Let [itex]\lambda[/itex] be an eigenvalue of [itex]A[/itex] such that
[tex](A - \lambda I)^n = 0[/tex]
but
[tex](A - \lambda I)^{m} \neq 0[/tex]
for every positive integer [itex]m < n[/itex].

Given that, can you show that [itex](A^T - \lambda I)^n = 0[/itex] and that there does not exist a positive integer [itex]m < n[/itex] such that [itex](A^T - \lambda I)^m = 0[/itex]?
 
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  • #3
catsarebad said:

Homework Statement


show that minimal poly for a sq matrix and its transpose is the same


Homework Equations






The Attempt at a Solution


no clue.

The polynomial ##p(x) = x^m + a_1 x^{m-1} + \cdots + a_{m-1} x + a_m ## is a minimal polynomial for matrix ##A## if and only if ##p(A) x = 0## for all column vectors ##x##, but this is not true for any polynomial of degree < m.

In other words, the vectors ##x, Ax, A^2 x, \ldots, A^{m-1}x## are linearly independent, but ##A^m x## is a linear combination of ##x, Ax, A^2 x, \ldots, A^{m-1}x##; furthermore, this same m and this same linear combination holds for all ##x##.

Basically, this is how some computer algebra packages find minimal polynomials, without finding the eigenvalues first. In fact, if we restrict the field of scalars to the reals a real matrix ##A## may not have (real) eigenvalues at all, but it will always have a real minimal polynomial.
 
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  • #4
pasmith said:
Let [itex]\lambda[/itex] be an eigenvalue of [itex]A[/itex] of geometric multiplicity [itex]n[/itex]. Then
[tex](A - \lambda I)^n = 0[/tex]
but
[tex](A - \lambda I)^{m} \neq 0[/tex]
for every positive integer [itex]m < n[/itex].

Given that, can you show that [itex](A^T - \lambda I)^n = 0[/itex] and that there does not exist a positive integer [itex]m < n[/itex] such that [itex](A^T - \lambda I)^m = 0[/itex]?

i'm not sure where we are going with this.

i assume this is a property
[tex](A - \lambda I)^n = 0[/tex]
but
[tex](A - \lambda I)^{m} \neq 0[/tex]
for every positive integer [itex]m < n[/itex].

but i don't get how showing the next part will help with minimal poly problem.
 
  • #5
catsarebad said:
i'm not sure where we are going with this.

i assume this is a property
[tex](A - \lambda I)^n = 0[/tex]
but
[tex](A - \lambda I)^{m} \neq 0[/tex]
for every positive integer [itex]m < n[/itex].

but i don't get how showing the next part will help with minimal poly problem.

Hint: [itex](A^n)^T = (A^T)^n[/itex]
 
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