Solving a System of Multiplicative Matrices

[electronicengi]

Please do not be offended by my literary style. I find thinking about mathematical problems in such a way helps me learn better.

A is a 2x2 matrix of complex numbers, call this "apple"
B is a 2x2 matrix of complex numbers, call this "banana"

Let a "Fruit Salad" be defined as follows:
S = AB

Suppose that I had two apples and two bananas in my fruit bowl
A1 A2 B1 B2

However, before I was able to measure my apples/bananas, my wife cut them up and mixed them to make four different servings of fruit salad:
S11 = A1B1
S12 = A1B2
S21 = A2B1
S22 = A2B2

So now I have a plethora of fruit salad {S11, S12, S21 and S22}, but I really want to know about the fruit that was put into it {A1, A2, B1 and B2}. How can I solve for A1, A2, B1 and B2 given S11, S12, S21 and S22?

That's the end of my question. The remainder of this post is to demonstrate my thoughts.

If my apples, bananas and fruit salads were scalar quantities I could set up the following simultaneous equations to solve via Gaussian elimination:
ln(S11) = ln(A1) + ln(B1)
ln(S12) = ln(A1) + ln(B2)
ln(S21) = ln(A2) + ln(B1)
ln(S22) = ln(A2) + ln(B2)

I am aware of the concept of a "matrix exponential" and its inverse, the "matrix logarithm". However, heading down this path looks like it is going to be messy. I thought I would first check to see if anyone knows of a nicer way to solve this problem.

 

[UltrafastPED]

Shouldn't S21 = A1B2 actually be A2B1?

 

[electronicengi]

You are correct UltrafastPED. I have modified the original post. My apologies.

 

[Office_Shredder]

I don't think this will work in general... e^A+B = e^Ae^B only if A and B commute, so I assume logarithm has the same property - that log(AB) = log(A) + log(B) only if A and B commute. You'll have to use the more complicated

http://en.wikipedia.org/wiki/Baker–Campbell–Hausdorff_formula

which doesn't really help since you don't get a set of linear equations

 

[electronicengi]

You raise a good point Office_Shredder.

I don't think I that my original question contains enough information to yield a unique solution. In the particular problem I am looking to solve A and B do not always commute. However, suppose that the following is known:
T11 = B1A1
T12 = B1A2
T21 = B2A1
T22 = B2A2

(That is, my wife made eight different types of fruit salad with the available fruit.)

 

[SteamKing]

Your 'fruit bowel' is actually a 'fruit bowl'. The 'bowel' is another name for the lower digestive tract.

 

[electronicengi]

Thank you Office_Shredder. Your comment about "eA+B = eAeB only if A and B commute" has highlighted something to me. My original question does not contain enough information to yield a unique solution. This is why an assumption (such as commutativity) is required for the mathematics to work out.

Suppose that
A = [ a₁₁ a₁₂ ; a₂₁ a₂₂ ]
B = [ b₁₁ b₁₂ ; b₂₁ b₂₂ ]
S = [ s₁₁ s₁₂ ; s₂₁ s₂₂ ]

In my original question I had assumed that I had provided 16 equations and 16 unknowns.

The unknowns were:
a₁₁, a₁₂, ...
b₁₁, ...

The equations were of the the form:
a₁₁b₁₁ + a₁₂b₂₁ = s₁₁
a₁₁b₁₂ + a₁₂b₂₂ = s₁₂
etc.

I was wrong in assuming this. What I had really constructed was a set of 16 linear equations with 32 unknowns, because the unknowns were actually of the following form:
a₁₁b₁₁, a₁₂b₂₁, a₁₁b₁₂, a₁₂b₂₂, ...

If I were to introduce some extra information into the question - For example, suppose my wife had made the following fruit salads in addition to the ones mentioned earlier:
T11 = B1A1
T12 = B1A2
T21 = B2A1
T22 = B2A2

(Note that assuming commutativity is the same as assuming Txy = Sxy where x,y = {1,2})

With this information I can construct a system of 32 linear equations with 32 unknowns. Gaussian elimination will enable me to solve for these 32 unknowns, which are of the form:
a₁₁b₁₁, a₁₂b₂₁, a₁₁b₁₂, a₁₂b₂₂, ...

I can then take the logarithm of each of these to set up another system of equations:
ln(a₁₁) + ln(b₁₁) = ln(a₁₁b₁₁)
ln(a₁₂) + ln(b₂₁) = ln(a₁₂b₂₁)
...

Following from this, I can use Gaussian elimination to solve for my unknowns:
ln(a₁₁)
ln(b₁₁)
ln(a₁₂)
ln(b₂₁)

From these terms I can calculate the following:
a₁₁, a₁₂, ...
b₁₁, ...

Hence solving for A1, A2, B1 and B2. (The components of my fruit salads.)

These are my thoughts so far. There may be an error somewhere in my line of reasoning, so I won't close the thread just yet. I shall go away to see if the extra information (i.e. T11, T12, T21 and T22) was what I needed to know to solve the problem.

 

[electronicengi]

Your 'fruit bowel' is actually a 'fruit bowl'. The 'bowel' is another name for the lower digestive tract.

How embarrassing. The typographical error has been corrected.

 

[SteamKing]

Seems like a lot of work, finding matrix exponentials and logarithms. Why don't you just multiply the coefficient matrices together?

It's not clear what the entries of A, B, and S are supposed to represent. Slices of apples, bananas and the number mixed together ...?

 

[electronicengi]

These are my thoughts so far. There may be an error somewhere in my line of reasoning, so I won't close the thread just yet. I shall go away to see if the extra information (i.e. T11, T12, T21 and T22) was what I needed to know to solve the problem.

Alas, no luck. I am not sure how to solve this one. For now I think I am going to resort to coding up an algorithm to find a numeric approximation.