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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 be an matrix such that where denotes the identity matrix. Show that is non-singular.

(b) Give an example of a non-zero real matrix such that for all real vectors .

Nilpotent Matrix

Eigenvalues

Skew-symmetric Matrix

The first part of the problem is quite easy,

It is given that for a matrix , we have , so, is a nilpotet matrix, right !

And we know that all the eigenvalues of a nilpotent matrix are . Hence all the eigenvalues of are 0.

Now let be the eigenvalues of the matrix . So, the eigenvalues of the nilpotent matrix are of form where, . Now since, which implies , for .

Since all the eigenvalues of are non-zero, infact . Hence our required propositon.

(b) Now this one is quite interesting,

If for any matrix, the Quadratic form of that matrix with respect to a vector is of form,

where and are the elements of the matrix. Now if we equate that with , what condition should it impose on and !! 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, is invertible right !! So, can you prove that we can always get two vectors from , say and , such that the necessary and sufficient condition for the invertiblity of the matrix is "** must be different from "** !!

This is a very important result for Statistics Students !! Keep thinking !!

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 be an matrix such that where denotes the identity matrix. Show that is non-singular.

(b) Give an example of a non-zero real matrix such that for all real vectors .

Nilpotent Matrix

Eigenvalues

Skew-symmetric Matrix

The first part of the problem is quite easy,

It is given that for a matrix , we have , so, is a nilpotet matrix, right !

And we know that all the eigenvalues of a nilpotent matrix are . Hence all the eigenvalues of are 0.

Now let be the eigenvalues of the matrix . So, the eigenvalues of the nilpotent matrix are of form where, . Now since, which implies , for .

Since all the eigenvalues of are non-zero, infact . Hence our required propositon.

(b) Now this one is quite interesting,

If for any matrix, the Quadratic form of that matrix with respect to a vector is of form,

where and are the elements of the matrix. Now if we equate that with , what condition should it impose on and !! 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, is invertible right !! So, can you prove that we can always get two vectors from , say and , such that the necessary and sufficient condition for the invertiblity of the matrix is "** must be different from "** !!

This is a very important result for Statistics Students !! Keep thinking !!

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