Matrix, minimal polynomial

  • #1

Homework Statement



A n × n-matrix A satisfies the equation A2 = A.

(a) List all possible characteristic polynomials of A.

(b) Show that A is similar to a diagonal matrix

Homework Equations





The Attempt at a Solution



A2 = A
so, A2 - A = 0
A(A-I) = 0

Our minimal polynomial is x2 - x = m(x)

Our eigenvalues are 0 and 1, and since in our minimal polynomial each one has a multiplicity of 1, A is similar to a diagonal matrix consisting of n jordan blocks with either 1's or 0's on the diagonal and 0 everywhere else since we have an nxn matrix and n jordan blocks.

So I've proved part (b) first.

For part (a) we don't know how many times each eigenvalue occurs in the characteristic polynomial, so p(t) = ta(t-1)b
where a,b=0,1,2,...,n.


Is this correct?
 
Last edited:

Answers and Replies

  • #2
22,129
3,297
All is correct, except

A(A-I) = 0

A = 0, or A = I

AB=0 does not imply A=0 or B=0. For example

[tex]\left(\begin{array}{cc} 1 & 0\\ 0 & 0\end{array}\right)[/tex]

is a matrix A that satisfied [itex]A^2=A[/itex], but which is not 0 or I.
 
  • #3
Okay, thanks! :smile:
 
  • #4
22,129
3,297
Hmm, couldn't the minimal polynomial also be a (polynomial) divider of x2-x?

:blushing: It's not my day today. Thanks ILS...
 
  • #5
I like Serena
Homework Helper
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:blushing: It's not my day today. Thanks ILS...

No, I just deleted my post.
They're asking for all characteristic polynomials, not for all minimal characteristic polynomials.
 
  • #6
22,129
3,297
No, I just deleted my post.
They're asking for all characteristic polynomials, not for all minimal characteristic polynomials.

No, you were correct. If a matrix satisfies [itex]A^2=A[/itex], then its minimal polynomial is not necessary [itex]x^2-x[/itex]. I should have pointed that out.

That said, this doesn't change anything about the solutions of the OP. They remain correct.
 
  • #7
HallsofIvy
Science Advisor
Homework Helper
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If A^2- A= A(A- I)= 0, then, whatever the characteristic equation of A is, it must include factors of x and x- 1.
 

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