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Matrix Powers - 2147483647 - 2011-05-22
Is there a specific, clean formula that I can use to raise a matrix to an arbitrary power? To keep this simple, let's use square, 2x2 matrices.
What is the "geometric" interpretation of doing such? I know that matrix multiplication serves to linearly distort spaces by changing the basis vectors, so repeatedly doing this would be repeatedly distorting the basis vectors by a certain amount. Thus, would raising a matrix to an arbitrary power distort the basis vectors to some kind of vector in between? Would each of the entries vary continuously?
The reason I ask is that I recently found out what a matrix exponential is:
e^[A] = [I] + [A] + [A][A]/2! + [A][A][A]/3! + ...
However, if I wanted to compute 10^[A], then I would have to rewrite it in terms of e. I can do this two ways:
10^[A] = (e^ln(10))^[A] = (e^[A])^ln(10)
or:
10^[A] = (e^ln(10))^[A] = e^([A]*ln(10))
This implies that:
([I] + [A] + [A][A]/2! + [A][A][A]/3! + ...)^ln(10)
= [I] + [A]*ln(10) + [A][A]*ln(10)^2/2! + [A][A][A]*ln(10)^3/3! + ...
Which doesn't really make sense to me.
Matrix Powers - Noah - 2011-05-22
There is a relatively simple way of raising a matrix to any power, even negative or decimal-powers:
Diagonalize A so that A = UPU⁻¹, and P is a diagonal matrix. Then Aⁿ = UPⁿU⁻¹ for any n. (I assume you know how to do this since you've been working with matrix exponents.)
For large matrices, you should not do this if you are not calculating a negative or decimal power, because the most efficient way of diagonalizing a matrix as of today is to, well, multiply it. You are basically doing the work twice. Of course, if you need several powers, diagonalizing is efficient.
And yeah, matrix exponential is
![[Image: 3ut2aft.png]](http://mathurl.com/3ut2aft.png)
,which is the exact same definition of e^x for any number x too, if you use Taylor series.
So, say we want to find 10^x. We can then express this in this way, as you did with the matrix:
![[Image: 3eb52g3.png]](http://mathurl.com/3eb52g3.png)
This means that
![[Image: 3aua6vs.png]](http://mathurl.com/3aua6vs.png)
So, as you see, there is nothing different about matrices, compared to real number when using this equation. Do you consider this weird, or is this just an unexpected result? Both sums converge, as k! grows faster than any polynomial or combination of them.
Noah
Matrix Powers - 2147483647 - 2011-05-22
Noah Wrote:Diagonalize A so that A = UPU⁻¹, and P is a diagonal matrix. Then Aⁿ = UPⁿU⁻¹ for any n. (I assume you know how to do this since you've been working with matrix exponents.)
There are several problems to this. I came across matrix exponentials as a way to solve first order linear systems (of ODEs). By diagonalizing the matrix, I'd have to compute its eigenvalues and eigenvectors, which then makes the process kind of useless because using the eigenvalues and eigvenvectors, I could compute the answer directly. Also, I'm not sure when this happens, but I do know that some matrices aren't diagonalizable, and I'm not sure whether or not all compositions (for lack of a better term) of n x 1 vectors into n x n matrices have inverses.
Noah Wrote:So, as you see, there is nothing different about matrices, compared to real number when using this equation. Do you consider this weird, or is this just an unexpected result? Both sums converge, as k! grows faster than any polynomial or combination of them.
It was an unexpected result, since I previously didn't think about diagonalization. I'm still not sure how this is fitting and if it varies continuously, since I lack results. One difficulty with that I face with this problem is that my graphing calculator can't handle matrix exponentials, non-integer powers, and negative powers, so I can't compute any results.
If we consider the identity matrix, the identity matrix to any power must equal itself. After all, that's the definition of the identity matrix. However, if we tried to "diagonalize it", we'd get:
Aⁿ = UPⁿU⁻¹
[1, 0; 0, 1]^n = [1, 0; 0, 1] [1, 0; 0, 1]^n [1, 0; 0, 1]
Or equivalently,
[1, 0; 0, 1]^n = [0, 1; 1, 0] [1, 0; 0, 1]^n [0, 1; 1, 0]
I'm not sure how this even changes anything.
What happens when the eigenvalues are complex? Is there a geometric interpretation of this? In general, is there anything useful that this is used for?
Matrix Powers - Noah - 2011-05-22
2147483647 Wrote:There are several problems to this. I came across matrix exponentials as a way to solve first order linear systems (of ODEs). By diagonalizing the matrix, I'd have to compute its eigenvalues and eigenvectors, which then makes the process kind of useless because using the eigenvalues and eigvenvectors, I could compute the answer directly. Also, I'm not sure when this happens, but I do know that some matrices aren't diagonalizable, and I'm not sure whether or not all compositions (for lack of a better term) of n x 1 vectors into n x n matrices have inverses.
Well, if you cannot diagonalizable the matrix, you will not have enough eigenvectors and -values. Therefore you cannot solve through eigenvalues and -vectors, and you need to either represent the problem in another way (probably without matrices), or use some other property to bypass the lack of eigenvectors and -values.
2147483647 Wrote:It was an unexpected result, since I previously didn't think about diagonalization. I'm still not sure how this is fitting and if it varies continuously, since I lack results. One difficulty with that I face with this problem is that my graphing calculator can't handle matrix exponentials, non-integer powers, and negative powers, so I can't compute any results.
Use Mathematica, Maple or Frink? Or Python, if you want to. (I usually use Python, actually).
2147483647 Wrote:If we consider the identity matrix, the identity matrix to any power must equal itself. After all, that's the definition of the identity matrix. However, if we tried to "diagonalize it", we'd get:
Aⁿ = UPⁿU⁻¹
[1, 0; 0, 1]^n = [1, 0; 0, 1] [1, 0; 0, 1]^n [1, 0; 0, 1]
Or equivalently,
[1, 0; 0, 1]^n = [0, 1; 1, 0] [1, 0; 0, 1]^n [0, 1; 1, 0]
I'm not sure how this even changes anything.
Is there any surprising effect here which surprises you? Of course the identity matrix of any power is the identity matrix.
2147483647 Wrote:What happens when the eigenvalues are complex? Is there a geometric interpretation of this? In general, is there anything useful that this is used for?
Oh, yeah, forgot to tell about that.
Nothing special, actually. Just work with it as you usually work with imaginary numbers.
I haven't really seen many visual examples of complex matrices. However, the eigenvalues of rotation-matrices are imaginary. In the 2D-plane, you can imagine that it works as magnetism, and that the eigenvectors and eigenvalues are the magnetic field which rotates the plane.
Noah
Matrix Powers - 2147483647 - 2011-05-22
Noah Wrote:Well, if you cannot diagonalizable the matrix, you will not have enough eigenvectors and -values. Therefore you cannot solve through eigenvalues and -vectors, and you need to either represent the problem in another way (probably without matrices), or use some other property to bypass the lack of eigenvectors and -values.
That's disheartening. How, then, do I raise a matrix with a repeated eigenvalue to an arbitrary power without the magical formula? ._.
Noah Wrote:Is there any surprising effect here which surprises you? Of course the identity matrix of any power is the identity matrix.
Ah. It's that earlier I didn't realize that raising a diagonal matrix to a power is equal to raising its terms to the same power, so I was fooling around with the identity matrix in hopes of getting insight as to how to do this.
Noah Wrote:Oh, yeah, forgot to tell about that.
Nothing special, actually. Just work with it as you usually work with imaginary numbers.
The thing is that raising a negative eigenvalue to an arbitrary power returns a complex value. This is strange behavior, since the powers would cause a sort of oscillation of the terms into the complex plane and then back to 0 at the integer powers, meaning that a completely real matrix could become complex in between powers. I'm not sure what this suggests, but this immediately makes me think that a matrix with complex terms can become real at certain non-integer powers. Then it follows that I should find the power that makes all the terms real? How should I go about doing that?
Noah Wrote:I haven't really seen many visual examples of complex matrices. However, the eigenvalues of rotation-matrices are imaginary. In the 2D-plane, you can imagine that it works as magnetism, and that the eigenvectors and eigenvalues are the magnetic field which rotates the plane.
Can you link me to such an image? I'm not sure I understand this, and Google isn't turning up anything.
Matrix Powers - hadriel - 2011-05-22
Wish I could help you, pity I'm not fond of matrices, plus my Math course (already the harder one) doesn't touch much on stuff like matrix exponentials. And I very much prefer doing matrices like crunching algebra.
I notice that matrices tend to have very few pictures, partly because it is but an operator [for the rotation case] so you can't exactly visualise an operator easily. Might be helpful though...
Hadriel
Matrix Powers - 2147483647 - 2011-05-22
hadriel Wrote:Wish I could help you, pity I'm not fond of matrices, plus my Math course (already the harder one) doesn't touch much on stuff like matrix exponentials. And I very much prefer doing matrices like crunching algebra.
I'm actually not doing this for any class. I'm just fooling around with it on my own out of curiosity. It's a shame your classes don't touch on them. I heard that the linear algebra class here doesn't touch on them either. But at least you should try to find out, since exponentiation is a basic operation, right?
This is all just related to my recent obsession with finding smooth extensions of basic operations into all real numbers, like using the gamma function to extend factorials and using fractional calculus to find a smooth continuation of differentiation and integration.
I wonder what I'll take interest in next. Hmm.
Matrix Powers - hadriel - 2011-05-22
More interested in molecular and cell biology, and chemistry, than Math =P
Strictly speaking, it's not a necessity for my class to teach that because it's not imperative, yet - I'm only in 1st year and we don't go that in depth. My pre-uni math education kinda neglected matrices, and I can survive without touching matrices e.g. doing Jacobian the algebric way instead of the matrix way (not that much diff... except in presentation). If they gave me exponentiation of matrices in exams, it's also not difficult to crunch it out - problem is that I won't really have much "geometric interpretation" of that.
I'm decent in Math at my level, don't really need too much more in Chemistry either... I use Math more as a tool than as a toy, but I'm not that curious about Math now. Some things are just NOT for curiosity when crunch time descends. [i.e. I'm having exams really soon, as are many people]
Hadriel
Matrix Powers - 2147483647 - 2011-05-22
[spoiler=Very off topic, so I'll put this in a spoiler.]
hadriel Wrote:e.g. doing Jacobian the algebric way instead of the matrix way (not that much diff... except in presentation)
How do you remember anything higher than a 2x2 Jacobian without a matrix?
hadriel Wrote:If they gave me exponentiation of matrices in exams, it's also not difficult to crunch it out - problem is that I won't really have much "geometric interpretation" of that.
I'm decent in Math at my level, don't really need too much more in Chemistry either... I use Math more as a tool than as a toy, but I'm not that curious about Math now. Some things are just NOT for curiosity when crunch time descends.
It's not like math is absolutely pointless and only used for toying. Also, in general, math shouldn't just be viewed as a just a subject that you learn so that you can take tests. A lot of math actually coincides with your interests:
1. Integrated rate laws. Some of them can be quite complicated, especially when you begin to consider more than one reactions, and decide not to neglect the "faster" reactions just because they don't seem to have an effect on the reaction rate during the middle of the reaction. For example, consider the elementary reaction:
2A -> 2B -> 2C
Rate laws are: (k1)[A]^2, and (k2)[A]^2
2. Modeling the temperature over time for very basic systems of two objects with different specific heat capacities heat using mc(dT) and Newton's Law of Cooling. Ie, two different materials with different heat capacities are brought into contact. Heat transfer happens at different rates, because the two materials are of different masses and specific heat capacities. Consider the following system:
dA/dt = -(Ma)(Ca)(k1)(A-B)
dB/dt = (Mb)(Cb)(k2)(A-B)
If A represents the temperature of the hotter material and B the colder, and k1 and k2 are adjustable constants, solve the system to get the temperature over time plot. The easiest/fastest way to do this learn how to use matrices and eigenvalues/eigenvectors to solve systems. And then you get a feel for what eigenvalues/eigenvectors actually are.
3. Not going to going to type out in detail here, but thermodynamics is full of calculus. Each differential in the equation:
dU = TdS - PdV
forms and defines the basic definitions of the quantity. For example, temperature is defined as dU/dS. Manipulating each quantity for different uses require certain calculus "tricks", such Legendre transformations, since the actual equations for each of the differentials are unknown.
hadriel Wrote:[i.e. I'm having exams really soon, as are many people]
Same here. In just over two weeks, actually.[/spoiler]
Matrix Powers - hadriel - 2011-05-23
Just a quick reply.
Spoiler
![[Image: morphine-gates-syn.gif]](http://www.chem.wisc.edu/areas/reich/syntheses/morphine-gates-syn.gif)
I'll attribute part of this to my lack of education in Physics. The rest of the sciences don't really go very intensively into Math... that is... until next year. I know the power of matrices but I really very much prefer other things like the spoiler.
Hadriel
Matrix Powers - Noah - 2011-06-01
2147483647 Wrote:Can you link me to such an image? I'm not sure I understand this, and Google isn't turning up anything.
I've looked after one, but I cannot find any good representation. You just need to multiply the U to the matrix A (image) you want to rotate, and visualize that one. So basically: Visualize A - it's a plane for our case. Then visualize AU, which is still a plane, but in three dimensions (i representing the z-coordinate). Then apply P^n, which does the "turning" of the AU-matrix, and then finally place it back in the normal plane by multiplying with U inverse.
It's hard to visualize operators, as hadriel said. Don't try too hard!
Noah
Matrix Powers - modular - 2011-06-01
Noah Wrote:I haven't really seen many visual examples of complex matrices. However, the eigenvalues of rotation-matrices are imaginary. In the 2D-plane, you can imagine that it works as magnetism, and that the eigenvectors and eigenvalues are the magnetic field which rotates the plane.
Noah
The only complex matrices that come readily to mind. Intro to quantum mechanics (junior in college level) will mention them, maybe. Grad level quantum might go more into them.