TIFR 2013 Problem 8 Solution - Diagonalizable Nilpotent Matrix

TIFR 2013 Problem 8 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India's premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.
The image is a front cover of a book named Introduction to Linear Algebra by Gilbert Strang. This book is very useful for the preparation of TIFR Entrance.

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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, $A=0$

you can also see the-TIFR 2014 Problem 11 Solution – Nilpotent Matrix Eigenvalues

Helpdesk

• What is this topic: Linear Algebra
• What are some of the associated concept: Invertible Matrix, Linear Transformation
• Book Suggestions: Introduction to Linear Algebra by Gilbert Strang

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