Correlation coefficient for weighted value pairs

I want to obtain an appropriate correlation statistic for sparse matrices, in two different cases.

Case 1

A small rectangular matrix (size: 3-20 elements long or wide) with a single value per row, per column. For example:

1  0  0
0  1  0
0  0  1   

0  1  0  0
1  0  0  0
0  0  1  0  => if rectangular, may have a col or row with all zeros.

0  1  0  0
0  0  1  0
1  0  0  0
0  0  0  1

My thinking here was to convert the values to paired coordinates. For the first matrix, we would obtain (1,1), (2,2), (3,3). Then pass these values into a linear correlation statistic like Pearson’s, which in this case would give a perfect negative correlation.

Is this the best approach?

Case 2

In second case, the values can be real valued, 0≤x≤1.

0.7  0    0
0    0.5  0
0    0    0.8

Higher values that are also more highly correlated should give a higher statistic than otherwise.

Does an appropriate correlation statistic exist for weighted value pairs like these? (1,1,w=0.7), (2,2.w=0.5), (3,3,w=0.8)