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.
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).
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
.
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.
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).
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
.
Suppose has two decompositions
where
and
and
and
.
Tberefore, .
But, we also have . Thus, by adding and subtracting, we get
.