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Kate2010
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Homework Statement
I'm working on a problem which is show that the matrix A = ([tex]^{a}_{c}[/tex] [tex]^{b}_{d}[/tex]) is diagonalisable iff (a-d)2 + 4bc > 0 or a-d = b = c = 0
Homework Equations
The Attempt at a Solution
I've shown if (a-d)2 + 4bc > 0 then the characteristic polynomial has 2 distinct roots so 2 distinct eigenvalues so is diagonalisable.
However, if (a-d)2 + 4bc [tex]\geq[/tex] 0 then it has a repeated root, so I have split it into cases:
If a-d = b = c then A is already diagonal
However, if a-d = b = 0 (c non-zero), the characteristic polynomial of A is already equal to 0. Does this mean A has no eigenvalues? Infinitely many? I will encounter a similar problem if a-d = c = 0 (b non-zero).
I am also unsure how to tackle (a-d)2 = 4bc.