INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More 

April 12, 2020

Tracing the Trace | ISI MStat 2016 PSB Problem 3

This ISI MStat 2016 problem is an application of the ideas of tracing the trace and Eigen values of a matrix and using a cute sum of squares identity.

Problem- Tracing the Trace

Suppose A is an \(n × n\) real symmetric matrix such that
\(Tr(A^2) = T r(A) = n\). Show that all the eigenvalues of A are equal to 1.

This problem is from ISI MStat 2016 PSB ( Problem #3)

Prerequisites

  • Trace of a Matrix
  • Eigen values of \(A^n\) w.r.t to the eigen values of \(A\).
  • Sum of Squares \(\geq 0\).

Solution

\( Av = {\lambda}v \Rightarrow A^nv = {\lambda}^nv\).

Since, A is a real symmetric matrix, then all the eigen values of the matrix A are real say {\( {\lambda}_1, {\lambda}_2, ..., {\lambda}_n\)}.

\(Tr(A^2) = \sum_{i=1}^{n} {{\lambda}_i}^2 = Tr(A) = \sum_{i=1}^{n} {{\lambda}_i} = n\)

\( \Rightarrow n\sum_{i=1}^{n} {{\lambda}_i}^2 = (\sum_{i=1}^{n} {{\lambda}_i})^2\)

\( \Rightarrow (n-1)\sum_{i=1}^{n} {{\lambda}_i}^2 = \sum_{i, j = 1, i \neq j }^{n} 2{\lambda}_i{\lambda}_j\)

\( \Rightarrow \sum_{i=1}^{n} ({{\lambda}_i - {\lambda}_j })^2 = 0\)

\( \Rightarrow {\lambda}_i = {\lambda}_j = \lambda \forall i \neq j \)

\( \Rightarrow Tr(A) = n\lambda = n \Rightarrow \lambda = 1\).

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com
enter