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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.

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.

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