This is a very simple sample problem from ISI MStat PSB 2009 Problem 1. It is based on basic properties of Nilpotent Matrices and Skew-symmetric Matrices. Try it !
(a) Let \(A\) be an \(n \times n\) matrix such that \((I+A)^4=O\) where \(I\) denotes the identity matrix. Show that \(A\) is non-singular.
(b) Give an example of a non-zero \(2 \times 2\) real matrix \(A\) such that \( \vec{x'}A \vec{x}=0\) for all real vectors \(\vec{x}\).
Nilpotent Matrix
Eigenvalues
Skew-symmetric Matrix
The first part of the problem is quite easy,
It is given that for a \(n \times n\) matrix \(A\), we have \((I+A)^4=O\), so, \(I+A\) is a nilpotet matrix, right !
And we know that all the eigenvalues of a nilpotent matrix are \(0\). Hence all the eigenvalues of \(I+A\) are 0.
Now let \(\lambda_1, \lambda_2,......,\lambda_k\) be the eigenvalues of the matrix \(A\). So, the eigenvalues of the nilpotent matrix \(I+A\) are of form \(1+\lambda_k\) where, \(k=1,2.....,n\). Now since, \(1+\lambda_k=0\) which implies \(\lambda_k=-1\), for \(k=1,2,...,n\).
Since all the eigenvalues of \(A\) are non-zero, infact \(|A|=(-1)^n \). Hence our required propositon.
(b) Now this one is quite interesting,
If for any \(2\times 2\) matrix, the Quadratic form of that matrix with respect to a vector \(\vec{x}=(x_1,x_2)^T\) is of form,
\(a{x_1}^2+ bx_1x_2+cx_2x_1+d{x_2}^2\) where \(a,b,c\) and \(d\) are the elements of the matrix. Now if we equate that with \(0\), what condition should it impose on \(a, b, c\) and \(d\) !! I leave it as an exercise for you to complete it. Also Try to generalize it you will end up with a nice result.
Now, extending the first part of the question, \(A\) is invertible right !! So, can you prove that we can always get two vectors from \(\mathbb{R}^n\), say \(\vec{x}\) and \(\vec{y}\), such that the necessary and sufficient condition for the invertiblity of the matrix \(A+\vec{x}\vec{y'}\) is " \(\vec{y'} A^{-1} \vec{x}\) must be different from \(1\)" !!
This is a very important result for Statistics Students !! Keep thinking !!
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