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18631377
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2013-09-05T08:28:38.313
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2013-09-06T08:54:44.310
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2013-09-06T08:54:44.310
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ok i try to write the problem as i see it: <pre><code>M = A x B </code></pre> <ul> <li>M known non negative integer input matrix NxN</li> <li>A,B unknown output decomposition of M, non unit, non negative integer</li> </ul> multiplication of matrixes: <pre><code>M[i][j] = sum(k=0,1,...,N-1)A[i][k]*B[k][j] </code></pre> ok now let me write an 3x3 example for clarity: <pre><code>M[3][3]=A*B i j i k k j i k k j i k k j M[0][0]=A[0][0]*B[0][0]+A[0][1]*B[1][0]+A[0][2]*B[2][0] M[0][1]=A[0][0]*B[0][1]+A[0][1]*B[1][1]+A[0][2]*B[2][1] M[0][2]=A[0][0]*B[0][2]+A[0][1]*B[1][2]+A[0][2]*B[2][2] M[1][0]=A[1][0]*B[0][0]+A[1][1]*B[1][0]+A[1][2]*B[2][0] M[1][1]=A[1][0]*B[0][1]+A[1][1]*B[1][1]+A[1][2]*B[2][1] M[1][2]=A[1][0]*B[0][2]+A[1][1]*B[1][2]+A[1][2]*B[2][2] M[2][0]=A[2][0]*B[0][0]+A[2][1]*B[1][0]+A[2][2]*B[2][0] M[2][1]=A[2][0]*B[0][1]+A[2][1]*B[1][1]+A[2][2]*B[2][1] M[2][2]=A[2][0]*B[0][2]+A[2][1]*B[1][2]+A[2][2]*B[2][2] // usage of B[i][j] M[0][0]=A[0][0]*B[0][0]+... M[1][0]=A[1][0]*B[0][0]+... M[2][0]=A[2][0]*B[0][0]+... M[?][j]=A[?][i]*B[i][j]+... // usage of A[i][j] M[0][0]=A[0][0]*B[0][0]+... M[0][1]=A[0][0]*B[0][1]+... M[0][2]=A[0][0]*B[0][2]+... M[i][?]=A[i][j]*B[j][?]+... </code></pre> When you look closer then the solution is very simple. find all: <pre><code>A[i][j]=GCD(M[i][0],...,M[i][N-1]) </code></pre> then derive B either from M,A or as B=M*inverse(A) M is decomposable if there is at least single A[i][j] > 1 also you can get B as GCD and derivate A from M,B,.... thats all , hope it helps... 
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singulars
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1. POundecomposability of a non-negative integer matrix
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1. USSpektre
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plurals
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