Question

Consider the matrica b d Assume llast (a.d = -4bc holds. is Ĥ always diagonalizable? If your answer is yes prove it! If your answer no give an example that Shows example llhat show A is not diagonalizable A is diagonaligable and give another

Answer 1

A IS NOT ALWAYS DIAGONALIZABLE.

Explanetion :-

A matrix is said to be diagonalizable if A=PDP^-1 where P is an invertible matrix and D is a diagonal matrix.

A matrix is diagonalizable if and only if all its eigen values are regular i.e. for all it's eigen values, algebraic multiplicity and geometric multiplicity is same.

Also we know that algebraic multiplicity ≥ geometric multiplicity.

Now,

a-2 oo Asla á] The charecterske folynomial of Am. det (A-2D) –0 (a-r) (d-a)- Be =0 >> at ad -bezo => ar-card) a tlad-be) 20 egen values are card) + Cata) -4 (ad-6c) dan ar card) a zz 2 az card) I & card)" 144d-4ad +46c - can y) } vě 4:45.mot named A has eigen value and with multiplicity 2

Two cases may appear now.

Case 1 :- Geometric multiplicity of .(a+d)/2 is 1.

IN THIS CASE A IS NOT DIAGONALIZABLE.

For example take $\small A=\begin{bmatrix} 2 & 1\\ 0& 2 \end{bmatrix}.$

Case 2 :- Geometric multiplicity of .(a+d)/2 is 1.

IN THIS CASE A IS DIAGONALIZABLE.

For example take $\small A=\begin{bmatrix} 2 & 0\\ 0& 2 \end{bmatrix}.$