complex matrices and diagonalisability (is that a word?? :P)

[Sarahisme]

hey,

 

i am not sure how to approach this problem.... any suggestions on how?

 

Thanks

 

Sarah

[Sarahisme]

anyone?

[Tom Mattson]

Start by writing down a general 2x2 upper triangular matrix:

 

[math]A=\left[\begin{array}{cc}a&b\\0&c\end{array}\right][/math]

 

where a,b,c are complex numbers.

 

Now, under what conditions is a matrix not diagonalizable?

[Sarahisme]

ok , yep i can do this bit, its the proof i ama having trouble with i think....

 

a non-diagonalisable 2x2 complex matrix is:

 

[math]A=\left[\begin{array}{cc}0&i\\0&0\end{array}\right][/math]

 

hows that?

 

however i don't know where to start with the proof... :S

[Tom Mattson]

When you're asked to prove things, you are supposed to identify what you can assume and then identify a goal.

 

Look at what you are being asked to do.

 

Prove that there exist an invertible matrix P such that B=P^-1AP is upper triangular.

 

Assume that P and A are 2x2 complex matrices, and that the product in the problem statement (which is equal to B) is upper triangular.

 

Prove that the matrix P exists and is invertible.

 

So start by writing down general matrices P=[p_ij], A=[a_ij], and B=[b_ij] (just remember that b₂₁=0). Do the multiplication, determine the p_ij in terms of the a_ij and b_ij, and show that the result is invertible.

[Sarahisme]

what does this notation mean?

 

"P=[p^ij[/sub]]"

[Tom Mattson]

That was a typo. It's fixed now.

[Sarahisme]

ok, its good, i've got it now! thanks for the help Tom