{"id":"matrix-rank","version":"1.0.0","description":"Compute the numerical rank of a finite-number matrix (square or rectangular).","supported_operations":["matrix rank","row rank","numerical rank"],"unsupported_operations":["exact symbolic rank over the rationals","nullspace basis computation"],"semantics":["Input matrix is a rectangular MxN array of finite JSON numbers, M and N each from 1 to 20, magnitude bounded to at most 1000000.","Computed via Gaussian elimination with partial (largest-magnitude) pivoting in IEEE-754 double precision, counting the number of pivot columns found.","Optional input tolerance (default 1e-9, range 0 to 1) is the minimum pivot magnitude treated as nonzero; smaller candidate pivots are treated as zero for rank purposes.","The result is an integer between 0 and min(M, N) inclusive."],"limits":{"max_rows":20,"max_cols":20,"max_abs_value":1000000},"pricing":{"status":"unpriced","charge_usd":null},"input_schema":{"type":"object","additionalProperties":false,"required":["matrix"],"properties":{"matrix":{"type":"array","minItems":1,"maxItems":20,"items":{"type":"array","items":{"type":"number"}}},"tolerance":{"type":"number","minimum":0,"maximum":1}}},"output_schema":{"type":"object","additionalProperties":false,"required":["result"],"properties":{"result":{"type":"integer"}}},"examples":[{"input":{"matrix":[[1,2],[2,4]]},"output":{"result":1}}],"execute_url":"/v1/tools/matrix-rank/versions/1.0.0/execute"}