Linesum

Linesum refers to the sum of elements that lie along a straight line within a grid-like array, such as a matrix or image. In many contexts, line sums are most familiar as row sums and column sums.

In a matrix A = [a_{i,j}] of size m × n, the row sum for row i is R_i = sum_{j=1}^n a_{i,j}, and the column sum for column j is C_j = sum_{i=1}^m a_{i,j}. The total sum of all entries is sum_{i=1}^m sum_{j=1}^n a_{i,j}. For square matrices, diagonal line sums are common, including the main diagonal S_main = sum_{i=1}^{min(m,n)} a_{i,i} and the anti-diagonal S_ant = sum_{i} a_{i,n-i+1} (where defined).

Line sums can also be defined along lines in directions other than rows and columns, or along lines in a discrete image. In image processing and tomography, the analogue is a projection or line integral: the sum of pixel values along a line through the image, which in continuous settings leads to the Radon transform.

Applications of line sums include data summarization in contingency tables, where row and column margins encode constraints; in linear programming and network flow, line sums serve as consistency checks for aggregated data. They also appear in puzzles such as Kakuro, where clues specify line sums for contiguous blocks.

Notes: the term linesum is used variably across disciplines. In formal contexts, it is usually clearer to specify row sum, column sum, or diagonal sum for precision.