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Calculating the pth power of an nxn matrix 219
Calculating the pth Power of an
nxn Matrix
It's finay time to
tackle today's real
problem! Finding the
pth power of an n×n
matrix.
eigenvalues and eigenvectors
of the matrix
8
2
−3
1
So let's just
build on that
example.
Makes
sense.
for simplicity’s
sake, Let's
chse
c
1
= c
2
= 1.
Using the two
calculations above...
Let’s multiply
to the right of both sides
of the equation. Refer to
page 91 to s why
exists.
220 Chapter 8 Eigenvalues and Eigenvectors
Try using
the formula
to calculate
2
8
2
−3
1
H...
okay.
Is...this it?
Yep!
Yay!
Lking at your
calculations,
would you say this
relationship might
betrue?
Uh...
It actuay is!
This formula is very useful
for calculating any power
of an n×n matrix that can be
wrien in this form.
Got it!
Oh, and by
the way...
When
p = 1
, we say that the formula
diagonalizes
the
n×n
matrix
a
11
a
21
a
n1
a
12
a
22
a
n2
a
1n
a
2n
a
nn
Nice!
And
that's it!
The eigenvector coesponding to
λ
1
The eigenvector coesponding to
λ
2
The eigenvector coesponding to
λ
n
The right side of the equation is the diagonalized
form of the mile matrix on the left side.
222 Chapter 8 Eigenvalues and Eigenvectors
That was the
lastleon!
How do you
fl? Did you
get the gist
of it?
Yeah, thanks
to you.
Awesome!
Reay, though,
thanks for helping
me out.
I know you're
busy,and you've
bn awfuy tired
because of your
karate practice.
Not at a! How
could I poibly
have bn tired
after a that
wonderful fd
you gave me?
I should
be thanking
you!
I' mi these
seions, you know!
My afternns wi
be so lonely from
now on...
We...we
could go out
sometime...
H?
Yeah...to lk
for math bks,
or something...
you know...
If you don't
have anything
else to do...
Sure,
sounds
like fun!
So when
would you
like to go?

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