Full-rank

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A matrix is full-rank when all of its rows are linearly independent. Otherwise, it is rank-deficient.

For example, the mapping matrix [41 65 95 115] 31 49 72 87] 19 30 44 53] is full-rank. We can check this by putting it into Hermite normal form (HNF), [1 0 0 -5] 0 1 0 2] 0 0 1 2], and observing that there are no rows of all zeros at the bottom (this is the mapping for marvel temperament). On the other hand, [41 65 95 115] 31 49 72 87] 10 16 23 28] is rank-deficient, because its HNF is [1 1 3 3] 0 6 -7 -2] 0 0 0 0], so we can see a row of all zeros has been produced at the bottom.

In Wolfram Language, an even quicker check for full-rank is possible, using MatrixRank[], which will give you the count of linearly independent rows of a matrix. If this is less than the count of rows, the matrix is rank-deficient.

You can guarantee a full-rank result by putting a matrix into canonical form.

One could generalize this notion to full-nullity and nullity-deficient when speaking of the linear independence of columns of a comma basis. And therefore further generalize the notion to grade, with full-grade and grade-deficient.

See also