how can i separate a matrix into a multiplication of two matrices One of Real Values and the Other for Integer values.

hello,

how can i separate a matrix X into a multiplication of two matrices Y and Z with Y contain Real Values and Z contain Integer values using MATLAB.

such as: X=Y*Z

with:

X= my Input Matrix of NxN.

and i search to obtain:

Y: matrix of integer values.

Z: Matrix of Real Values.

X is suposed to be a NxN matrix.

Answer 1

[Bruno Luong]

Of course.

Test matrix

X = rand(5)
n = length(X)

then

Z = ceil(1000*rand(n,n))
Y = X/Z
Y*Z

[ahcen aliouat]

great,

thank you very much Bruno!

[ahcen aliouat]

hello,

please @bruno could you give me an idea on how to get it in a way that the Real values matrix will be a diagonal matrix? ( identified just in the diagonal )

i will appreciate your help.

thank you

[Bruno Luong]

In general it's not possible for any X.

If you have such decompistion, it's equivalent to each column of X has a common divisor (on the diagonal of Z). And that is right for all column.

This condition is quite strict. But if you know X meets this, then you can write down:

X(:,j) = sum k_(i,j) d_j, for all j=1,2,...,n, with k_(i,j) integers

Then simply take

Z = diag(d)
Y(i,j) = k_(i,j)

You might try to apply euclidian algorithm to find a common divisor of floating points (if any), I'm not sure the bad-effect of floating points round-off on such algorithm (I guess it's far from robust).

Answer 2

[Raj]

There are infinite combinations possible (atleast by theory) in which you can split a real number to multiplication of a real number and an integer. Let me explain:

X=rand(N,N) % your matrix
Y=randi(M,N) % any NxN integer matrix with upper limit as M. INFINITE Possibilities here.

then

Z=X./Y % matrix of real values.

So that you can multiply Y & Z to get back X

X1=Y.*Z
isequal(X,X1) % confirms that X and X1 are equal.

Cheers!!

[ahcen aliouat]

thank you very much.

do you have an idea on how to get it in a way that the Real values matrix will be a diagonal matrix? ( identified just in the diagonal )

i will appreciate your help.

thank you