Point P partitions the directed line segment from A to B into the ratio 3:4. Will P be closer to A or B? Why
2 answers:
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Answer:
(a) Yes,
(b) Yes,
Step-by-step explanation:
First, lets understand what are eigenvectors and eigenvalues?
Note: I am using the notation to denote Lambda(A) sign.
is an eigenvector of matrix A with eigenvalue
is also eigenvector of matrix B with eigenvalue
So we can write this in equation form as
So what does this equation say?
When you multiply any vector by A they do change their direction. any vector that is in the same direction as of , then this
is called the eigenvector of
.
is
times the original
. The number
is the eigenvalue of A.
this number is very important and tells us what is happening when we multiply
. Is it shrinking or expanding or reversed or something else?
It tells us everything we need to know!
Bonus:
By the way you can find out the eigenvalue of by using the following equation:
where I is identity matrix of the size of same as A.
Now lets come to the solution!
(a) Show that is an eigenvector of
and find its associated eigenvalue.
The eigenvalues of and
are
and
, then
so,
which means that is also an eigenvector of
and the associated eigenvalues are
(b) Show that is an eigenvector of
and find its associated eigenvalue.
The eigenvalues of and
are
and
, then
so,
which means that is also an eigenvector of
and the associated eigenvalues are