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# The Unique Decomposition | ISI MStat 2015 PSB Problem 3 The solution plays with Eigen values and vectors to solve this cute and easy problem in Linear Algebra from the ISI MStat 2015 problem 3.

## Problem

Let be a real valued and symmetric matrix with entries such that and .
(a) Prove that there exist non-zero column vectors and such that and .
(b) Prove that every vector has a unique decomposition where and .

This problem is from ISI MStat 2015 PSB ( Problem #3).

### Prerequisites

• Eigen values and Eigen vectors

## (a)

Let's say is an eigenvalue of . Let's explore the possibilities of . . Since, is arbitrary, we get .

Since is real symmetric, it has real eigenvalues, and the possibilities are 1 and -1. Since, , there exists non-zero column vectors and such that and .

## (b)

Suppose has two decompositions where and and and .

Tberefore, .

But, we also have . Thus, by adding and subtracting, we get .

The solution plays with Eigen values and vectors to solve this cute and easy problem in Linear Algebra from the ISI MStat 2015 problem 3.

## Problem

Let be a real valued and symmetric matrix with entries such that and .
(a) Prove that there exist non-zero column vectors and such that and .
(b) Prove that every vector has a unique decomposition where and .

This problem is from ISI MStat 2015 PSB ( Problem #3).

### Prerequisites

• Eigen values and Eigen vectors

## (a)

Let's say is an eigenvalue of . Let's explore the possibilities of . . Since, is arbitrary, we get .

Since is real symmetric, it has real eigenvalues, and the possibilities are 1 and -1. Since, , there exists non-zero column vectors and such that and .

## (b)

Suppose has two decompositions where and and and .

Tberefore, .

But, we also have . Thus, by adding and subtracting, we get .

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