I hope this helps :)What is the expression for the matrix M^n in terms of n?

[agary12]

Homework Statement

I'm in 11th grade and I've been given the following in a series of problems:

(2 0)
(0 2)
Calculate M^N for 1,2,3,4,5,10,20,50. Describe any patterns you observe. Generalize the pattern into an expression for the matrix M^n in terms of n.

Homework Equations

The Attempt at a Solution

(2 0)
(0 2)^2 =

(4 0)
(0 4)

(2 0)
(0 2)^3 =

(8 0)
(0 8)

(2 0)
(0 2)^4 =

(16 0)
(0 16)

((2 0)
(0 2)^5 =

(32 0)
(0 32)

(2 0)
(0 2)^10 =

(1024 0)
(0 1024)

(2 0)
(0 2)^20 =

(1048576 0)
(0 1048576)

It looks like to me that you can multiply the value in the prior matrix by 2 (for powers 1-5) to get the new value in the next one. For example:

(2 0) (16 0)
(0 2)^4 = (0 16) so multiply 16 by 2 and you have 32. You then know that the matrix to the power of 5 will look like this:
(32 0)
(0 32)

Can someone help me find a rule in terms of n for M^n?

 

[tiny-tim]

Welcome to PF!

Hi agary12! Welcome to PF!

It looks like to me that you can multiply the value in the prior matrix by 2 (for powers 1-5) to get the new value in the next one. For example:

(2 0) (16 0)
(0 2)^4 = (0 16) so multiply 16 by 2 and you have 32. You then know that the matrix to the power of 5 will look like this:
(32 0)
(0 32)

Can someone help me find a rule in terms of n for M^n?

Well, you're almost there …

if M^N =

(k_N 0)
(0 k_N)

what is the rule for finding k_N?

 

[agary12]

I have a question about what you said,

M^N =
(kN 0)
(0 kN)

Is K being multiplied by N or is it just being included to show k is effected by N?

using this particular example:

(2 0)
(0 2)^3 =

(8 0)
(0 8)

What would the K value even be for the above value?
To get 8 from 2 you have to put it to the 3rd power, but that doesn't give me any new information. You could also multiply it by 4, but there is no 4 in the problem.

 

[lanedance]

i think tiny tim is saying try to find;

k(n) = some function n...

can you describe k(n)? (i think you pretty much described it in your last post...)

 

[tiny-tim]

Is K being multiplied by N or is it just being included to show k is effected by N?

It's just an index

(btw, try using the X² and X₂ tags just above the Reply box)

using this particular example:

(2 0)
(0 2)^3 =

(8 0)
(0 8)

What would the K value even be for the above value?
To get 8 from 2 you have to put it to the 3rd power, but that doesn't give me any new information. You could also multiply it by 4, but there is no 4 in the problem.

ok, that's the N = 3 case …

how about N = 4 … what's the pattern, and the mathematical rule of that pattern?

 

[rock.freak667]

(2 0)
(0 2)
= 2* (1 0)
(0 1)

What does this matrix represent? Now what is any matrix mulitiplied by this matrix?

 

[HallsofIvy]

I have a question about what you said,

Is K being multiplied by N or is it just being included to show k is effected by N?

using this particular example:

(2 0)
(0 2)^3 =

(8 0)
(0 8)

What would the K value even be for the above value?
To get 8 from 2 you have to put it to the 3rd power, but that doesn't give me any new information. You could also multiply it by 4, but there is no 4 in the problem.

He is suggesting that you look at the numbers 2, 4, 8, 16, 32, 64, 128, etc., which are what you get with n= 1, 2, 3, 4, 5, 6, 7, etc. What function of n are those?

rock.freak667's suggestion, that you look at powers of $2\begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}$ is also good.

 

[agary12]

I'm still confused by what rock freak is trying to say. I understand that this:
(2 0)
(0 2)
is equal to:
2*(1 0)
(0 1)
but why is that even important? Are you suggesting I do something with this matrix?
OH, I just typed it into my calculator and I think I may have found something. Basically this is just a
(1 0)
(0 1) matrix multiplied by 2
The only thing that you are doing when you add an exponent is putting the 2 to a power, at which point it is distributed into the matrix right?

$$\left\lfloor$$

He is suggesting that you look at the numbers 2, 4, 8, 16, 32, 64, 128, etc., which are what you get with n= 1, 2, 3, 4, 5, 6, 7, etc. What function of n are those?

K= 2, 4, 8, 16, 32, 64, 128
N= 1, 2, 3, 4, 5, 6, 7

Sorry but I'm not sure what you mena by "what function of n are those," are you saying that I need to find what is done to N to get K? In that case I am not sure. Nothing is being consistently multiplied by the N value to get K (N * X =/= k) since:
2/1 = 2
4/2 = 2
8/3 = 2.66
16/4 = 4
32/5 = 6.4
so there is no relationship found doing what I just did. I'm not sure what else I could do to N to get K.

 

[lanedance]

there's nothing say its a linear relation with n, and in fact its clearly not

you've said it a few times in the post so how about looking again at $$2^n$$ ?

 

[agary12]

there's nothing say its a linear relation with n, and in fact its clearly not

you've said it a few times in the post so how about looking again at LaTeX Code: 2^n ?

Why do you mean by looking at 2^N?

Basically I've found that the 2 values other than zero in M^n are found by putting 2 to the power you are putting the overall matrix to.

Therefore in this matrix
(2 0) LaTeX Code:^5
(0 2)

I can find the new values by taking 2^5= 32
The new matrix therefore is
(32 0)
(0 32)

Is the rule then just 2^n to find the new values within the matrix? I'm not sure if this is what they are looking for.

In response to tiny tim:

ok, that's the N = 3 case …

how about N = 4 … what's the pattern, and the mathematical rule of that pattern?

See this is the problem, I understand that you are putting 2^3 to get 8 and that 2^4 is 16, but I'm not sure what the pattern is that they are looking for. They are simply powers of 2, but how can I say this mathematically? And what do you mean by the "mathematical rule of that pattern"?

 

[lanedance]

write down M as a function of n. I think what they want is M(n)

you've pretty much told us in previous posts what this is. M(n) is a multiple of the identity matrix, with scalar multiplier $$2^n$$

this should be enough for what you're trying to do, but for a more general case you could write each specific element of M, to do this think about each element of M
$$m_{ij}(n)$$, where i = row, j = column

note m is diagonal, and multiply of the identity

so what is $$m_{ij}(n)$$
when $$i = j$$?
and when $$i \neq j$$?

in short i think you've got everything you need, just have to pull it together...

 

[tiny-tim]

Hi agary12!

just got up :zzz: …

See this is the problem, I understand that you are putting 2^3 to get 8 and that 2^4 is 16, but I'm not sure what the pattern is that they are looking for. They are simply powers of 2, but how can I say this mathematically?

oh i see!

you've got it, but you think you haven't!

how can you say mathematically "They are simply powers of 2"? …

you say a_N = 2^N

ok, so what is M^N ?

 

[agary12]

M^N equals:
(2^N 0^N)
(0^N 2^N)
Is that it?

Sorry lane dance, I'm not sure what your saying with the i and j subscripts. I haven't ever seen those before.

 

[tiny-tim]

M^N equals:
(2^N 0^N)
(0^N 2^N)
Is that it?

Yup!

… except of course please write 0 not 0^N

(btw, rock.freak667 was saying that M = 2I (where I is the unit matrix), so M^N = 2^NI^N = 2^NI )

 

[lanedance]

no worries agary12, look like you've got there, good worki

If you find below confusing, don't worry about it for now, but thought I'll just add a bit for completeness:

the i subscript relates to the row (horizontal line across matrix), so i = 1 is teh first row, i = 2 is the 2nd row & so on. Simialrly the j relates to the column (vertical line down matrix)

So each element of teh matrix is identifed uniquely by a single i,j reference

They don't have to be the letters i & j, its just a reference, could be any letter, but you see i & j quite often in books

in terms of the identity matrix, it is often written
$$\textbf{I}=$$

$$(1, 0)$$
$$(0, 1)$$

=
$$(\delta_{11}, \delta_{12})$$
$$(\delta_{21}, \delta_{22})$$

where $$\delta_{ij}$$ is the kronecker delta defined by:
$$\delta_{ij} = 1$$, if $$i = j$$
$$\delta_{ij} = 0$$, if $$i \neq j$$

in terms of your matrix it would look like
$$\textbf{M}(n) =$$

$$(m_{11}, m_{12})$$
$$(m_{21}, m_{22})$$

=
$$(2^n,0)$$
$$(0,2^n)$$

so you could write your formula for M as
$$m_{ij} = 2^n\delta_{ij}$$

this is equivalent to

M = 2ⁿ I