(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that A=[a,b;c,d]

is diagonalizable if -4bc < (a-d)^2

is not diagonalizale if -4bc > (a-d)^2

2. Relevant equations

For an nxn matrix, if there are n distinct eigenvalues then the matrix is diagonalizable.

For an nxn matrix, if there are n linearly independent eigenvetors then the matrix is diagonalizable.

3. The attempt at a solution

Characteristic equation for eigenvalues:

|λ-a,-b;-c,λ-d| = λ^2 + λ(a-d) + (ab - bc) = 0

λ = 0.5 * (d - a plus-or-minus sqrt((a-d)^2 - 4ad + 4bc))

For part 1, I need to show that there are two distinct solutions to this quadratic equation (b^2 - 4ac > 0: a,b,c from general quadratic eq - not from this problem)

I know that (a-d)^2 + 4bc > 0

However, I do not know how to show that (a-d)^2 - 4ad + 4bc > 0

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# Homework Help: Linear Algebra Homework Question

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