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# Eigen Values | ISI MStat 2019 PSB Problem 2 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.

## Problem

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 .

### Prerequisites

• Trace of a Matrix
• AM - GM Inequality
• Pigeon Hole Principle: Let the average of a set of positive numbers be . Use the pigeonhole principle to show that there exists at least one number less than or equal to .
• Algebra of Diagonal Matrices
• Eigenvalues of a Diagonal Matrix are the diagonal elements.

## Solution

(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.

## Problem

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 .

### Prerequisites

• Trace of a Matrix
• AM - GM Inequality
• Pigeon Hole Principle: Let the average of a set of positive numbers be . Use the pigeonhole principle to show that there exists at least one number less than or equal to .
• Algebra of Diagonal Matrices
• Eigenvalues of a Diagonal Matrix are the diagonal elements.

## Solution

(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!

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### One comment on “Eigen Values | ISI MStat 2019 PSB Problem 2”

1. Akshay says:

We have first prove the eigenvalues of A+B and A-B are real. Also, there are examples online contradicting the first to statements.

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