>> Let's talk about
similar matrices.
So first a definition.
To matrices, A and B, suppose
they are square matrices.
A is said to be similar to B if
there's an invertible matrix P
that has the property
that A is equal
to P times B times P inverse.
So some things to notice
here in the definition.
We are only defining
this property to talk
about square matrices and both
have to be N by N same size.
And the definition is
inexistent statement.
And so to matrices are similar
if there exists this
invertible matrix P
that has a certain property.
This matrix P if it exists is
also going to be the same size
as the matrices A and B. And
notice, if A is similar to B,
so that A is P times
B times P inverse.
Then we can solve for B and C
that B is P inverse times A
times P. What that means is
that we can write
B as Q A Q inverse
where Q is just P inverse.
In other words, if A is similar
to B then B is similar to A.
So this similarity
property is symmetric.
So next is a theorem that
talks about similar matrices
and its relationship
to igon values.
So two N by N matrices A
and B, if they are similar,
then they have the same
characteristic polynomial.
And that tells us
that they are going
to both have the
same igon values
with the same multiplicities.
So here's a quick proof.
We just need to show that the
characteristic polynomial each
one-- they characteristic
polynomial of A is equal
to the characteristic polynomial
of B whenever A and
B are similar.
And so we are just going
to compute using the fact
that A is similar to B. And
properties of the determinant.
So if A is similar to B then
there is this invertible matrix
P so that A is P times
B times P inverse.
And then A minus lambda I
we can write it as P, B,
P inverse minus lambda I.
And the identity matrix
is just P times P inverse.
And so we can write
this as P, B,
P inverse minus lambda
P, P inverse.
And now we can factor out P
from the right and from the left
and get that this is P times the
quantity B minus lambda I times
P inverse.
And then we just
take determinant.
So the determinant
of A minus lambda I,
this is the characteristic
polynomial for A. This is equal
to the determinant of P times
the quantity B minus lambda I
times P inverse and
now we are going
to use the multiplicative
property of determinant.
So that's determinant P times
the characteristic polynomial
of B times determinant
P inverse.
And determinant P
inverse remember,
is just 1 over determinant
P. And so those are scalars
that cancel and therefore
determinant A is determinant A
minus lambda I is
determinant B minus lambda I.
So this shows that the
characteristic polynomials are
the same.
So here's a warning, though.
So similarity is not the
same as row equivalents.
When you do row operations
usually you are going
to change the igon
values of a matrix.
The second warning
is similarity implies
that matrices have
the same igon values
but having the same igon values
does not imply the matrices
are similar.
So for example, this matrix 5,
3, 0, 5 has the same igon value
as 5, 0, 0, 5 but the
matrices are not similar.
Later, what we're going
to see is similar matrices
represent the same linear
transformation but just in
terms of a different basis.
So a different choice
of coordinate system.
This will end up
being very useful
when we start changing
coordinates in order
to study systems of linear
equations and we are going
to find coordinates that
are suitably adapted
to solving the problems
we are interested in.
