Expert: Scotto - 11/7/2008

Question
QUESTION: Dear Sir,
Attached the image of matrix equation. I have to develop computer program for that.Could you please tell me how to simplify it in order to develop the computer program?
Thanks in advance.
Kumaran

ANSWER: I'm not sure what bias is, but I'll call it ß to go with α.  The right hand side looks like a µ.

This simplifies to the equations
α(k+1) = α(k) - dt*ß(t) + dt*µ(k) and
ß(t+1)= ß(t).

From what you've given, ß never changes.  All you need to do is have a variable α which is either a single variable (if you only care about what it is currently equal to) or an array of numbers if you care about the value of it at each t.

The variable dt can be treated like a constant along with the variable µ(k) from what is shown here.  It also look like the variable ß(t) will also be a constant by what was given.


---------- FOLLOW-UP ----------

QUESTION: The matrix image is attached once again

The reason that I need to know how it was simplified is that there are other matrices with similar properties that need to be simplified?

Answer
I'm not sure this is what you want, but to multiply matrices together, multiply each row of the matrix my the each element in the vector.  First element in row 1*first element in vector + second element in row 1 * second element in vector and that gives the resulting element 1 in the product vector.  To get the 2nd elements in the product vector, do the same thing with the second row of the matrix.

Is that what you need to know?

So if you are multiplying a 3x3 matrix by a 3x1 vector, take each row of the matrix times the corrseponding row in the vector to get the row in the resulting vector.

Example
3 4 5
6 7 8
2 3 4
times the vector
44
-22
33

result
3*44 + 4(-22) + 5*33 = 209

Is this what you were after?
6*44 + 7(-22) + 8*44 = 462
2*44 + 3(-22) + 4*33 = 154

the result would be the vector
209
462
154

Note that whenever a matrix and a vector are multiplied together, the number of columns in the matrix must be the same as the number of rows in the vector.  To multiply two matrices, they also need the columns in the first matrix to be the same as the rows in the second matrix.  To get element i,j in the resulting matrix, take each element in row i of the first matrix times each element in column j of the second matrix, adding them up as you go.  As was said, this gives element i,j in the resulting matrix.

Note that multiplication is not the same in reverse order like it is in real numbers.  For two matrices A and B, AB is not equal to BA (except in a few special cases where chance is in your favor).

Not only are they different, but the may be different sizes.  If A were a 3*2 and B were a 2*3, AB would be 3x3 and BA would be 2x2.