This post based on eigen values of matrices and using very basic inequalities gives a detailed solution to ISI M.Stat 2019 PSB Problem 2.
Let and
be
matrices. Suppose that
has eigenvalues
and
has eigenvalues
where each
(a) Prove that has at least one eigenvalue greater than 2.
(b) Prove that has at least one eigenvalue greater than 0.
(c) Give an example of and
so that 1 is not an eigenvalue of
.
(a)
Consider the trace of .
Mean of the eigen values of +
=
=
=
=
. [ Hint: AM - GM Inequality ].
Now, use Pegion Hole Principle as mentioned in the prerequisites.
(b)
Mean of the eigen values of -
=
=
=
=
. [ Hint:
.]
Now, use Pegion Hole Principle as mentioned in the prerequisites.
(c)
Let's take and
. Now observe that
and
satisfy the given conditions.
. But
has an eigenvalue 1.
So, what to do? We want none of the diagonal values of to be not 1.
Take, and
. ow observe that
and
satisfy the given conditions.
, which has no eigen value 1.
Stay tuned!
This post based on eigen values of matrices and using very basic inequalities gives a detailed solution to ISI M.Stat 2019 PSB Problem 2.
Let and
be
matrices. Suppose that
has eigenvalues
and
has eigenvalues
where each
(a) Prove that has at least one eigenvalue greater than 2.
(b) Prove that has at least one eigenvalue greater than 0.
(c) Give an example of and
so that 1 is not an eigenvalue of
.
(a)
Consider the trace of .
Mean of the eigen values of +
=
=
=
=
. [ Hint: AM - GM Inequality ].
Now, use Pegion Hole Principle as mentioned in the prerequisites.
(b)
Mean of the eigen values of -
=
=
=
=
. [ Hint:
.]
Now, use Pegion Hole Principle as mentioned in the prerequisites.
(c)
Let's take and
. Now observe that
and
satisfy the given conditions.
. But
has an eigenvalue 1.
So, what to do? We want none of the diagonal values of to be not 1.
Take, and
. ow observe that
and
satisfy the given conditions.
, which has no eigen value 1.
Stay tuned!
We have first prove the eigenvalues of A+B and A-B are real. Also, there are examples online contradicting the first to statements.