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Prove that the matrix
A=
a b
c d
is diagonalizable if −4bc < (a − d)2 and is not diagonalizable if −4bc > (a − d)2
P roof :
To prove that the matrix A is diagonalizable, we will use a theorem that if an n × n matrix A has n
distinct eigenvalues, then A is diagonalizable
...
So by appyling the quadratic formula, we get the roots of the characteristic polynomial of A
...
e
...
e
...
So, A is diagonalizable if −4bc < (a − d)2 while if
(a − d)2 < −4bc then the characteristic polynomial has imaginary roots and A is not diagonalizable