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.

We've already found the

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

chse

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!

Lking at your

calculations,

would you say this

relationship might

betrue?

Uh...

It actuay is!

This formula is very useful

for calculating any power

of an n×n matrix that can be

wrien 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 coesponding to

λ

1

The eigenvector coesponding to

λ

2

The eigenvector coesponding to

λ

n

The right side of the equation is the diagonalized

form of the mile matrix on the left side.

222 Chapter 8 Eigenvalues and Eigenvectors

That was the

lastleon!

How do you

fl? Did you

get the gist

of it?

Yeah, thanks

to you.

Awesome!

Reay, though,

thanks for helping

me out.

I know you're

busy,and you've

bn awfuy tired

because of your

karate practice.

Not at a! How

could I poibly

have bn tired

after a that

wonderful fd

you gave me?

I should

be thanking

you!

I' mi these

seions, you know!

My afternns wi

be so lonely from

now on...

We...we

could go out

sometime...

H?

Yeah...to lk

for math bks,

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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