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Decomposition of Matrix

 

 

Decomposition is a method of splitting a matrix into multiplication of multiple matrix in the following form.

 

M = AB..N

 

How many matrix you split M into and the characteristics of splitted matrix varies depending on each specific decomposition method.

Actually 'breaking one thing into multiple other things' is one of the most common techniques in mathematics. For example, we often break a number into multiples of other numbers as follows.

 

15 = 2 x 3 x 5

 

We also break a polynomial into the multiples of other polynomials as shown below.

 

x^4 - 5 x^3 - 7 x^2 + 29 x + 30 = (x+2)(x+1)(x-3)(x-5)

 

Question is "Why we do this kind of break down ?". Sometimes we may do this kind of thing just for mathematical fun or curiosity, but in most case (especially in engineering area) we do this because we can get some benefit from it. For example, if you are asked to plot a graph for x^4 - 5 x^3 - 7 x^2 + 29 x + 30, you may have to do a lot of work (a lot of punching keys on your pocket calculator), but if you break the polynomial into (x+2)(x+1)(x-3)(x-5), you would be able to overal shape of the graph without doing even single calculation. Samething applies to Matrix decomposition. We decompose a Matrix into multiple other matrices because there are advantages doing it. What kind of benefit you can get from the matrix decomposition ? We can think of mainly two advantages

  • it helps getting solution for a matrix equation with very large number of elements
  • it helps you understand the characteristics of the original matrix (or original system equation that the matrix represents)

Then you may ask "Why I don't see this kind of benefit in the linear algebra course ?". "It just look like trying to make simple things more complex.", "It is seems to be designed just to give headache to students".

Mainly two reasons for this

  • Linear Algebra textbook or the course hardly mention the practical meaning and advanges of them. They just shows calculation process.
  • Most of textbook examples are very small sized matrix from a small set of system equations. In this case, the overhead of decomposing the matrix is greater than the benefit you can get from the process. So you wouldn't see any benefit from it and keep grumping "Why do I have to do this kind of meaningless stuffs".

However if you are given a matrix equation with a huge matrix (like 1000000 x 1000000), then you would start seeing the benefit of doing decomposition. Of course, you cannot decompose 1000000 x 1000000 by hand. You have to use computer. Then you may ask "If I use computer, why bother to decompose ? Computer would do the calculation directly from the original matrix". But in reality it is not. If the size of matrix is very large, there would be huge differences between with and without decomposition even for the high performance computer.

 

See http://en.wikipedia.org/wiki/Matrix_decomposition and see followings in sharetechnote in this page as a specific example. (I will keep adding more examples when I have chance)