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Diagonalizable Nilpotent Matrix (TIFR 2013 problem 8)

Question:

True/False?

If a real square matrix \(A\) is similar to a diagonal matrix and satifies \(A^n=0\) for some \(n\), then \(A\) must be the zero matrix.

Hint: There exists an invertible matrix \(P\) and a diagonal matrix \(D\) which satisfies \(PDP^{-1}=A\). What happens when we apply the given condition?

Discussion:

\(0=A^n=(PDP^{-1})^n=PDP^{-1}PDP^{-1}…PDP^{-1} \) (n-times multiplication)

Hence, \(0=PD^nP^{-1}\). \(P\) being invertible, we multiply on left and right by \(P^{-1}\) and \(P\) respectively and get \(D=0\).

In whatever basis you write the zero transformation, the result is same, namely the matrix of zero transformation is always zero-or null matrix.

Hence, \(A=0\)

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