Full-rank: Difference between revisions
Cmloegcmluin (talk | contribs) update EBK style |
Cmloegcmluin (talk | contribs) correct article; no such thing as full-grade, full-nullity, grade-deficient, or nullity-deficient |
||
| Line 1: | Line 1: | ||
A matrix is '''full-rank''' when all of its rows are [[linearly independent]]. Otherwise, it is '''rank-deficient'''. | A matrix is '''full-rank''' when either all of its rows are [[linearly independent]] or all of its columns are linearly independent. Otherwise, it is '''rank-deficient'''. | ||
For example, the [[mapping]] matrix {{rket|{{map|41 65 95 115}} {{map|31 49 72 87}} {{map|19 30 44 53}}}} is full-rank. We can check this by putting it into [[Hermite normal form]] (HNF), {{rket|{{map|1 0 0 -5}} {{map|0 1 0 2}} {{map|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, {{rket|{{map|41 65 95 115}} {{map|31 49 72 87}} {{map|10 16 23 28}}}} is rank-deficient, because its HNF is {{rket|{{map|1 1 3 3}} {{map|0 6 -7 -2}} {{map|0 0 0 0}}}}, so we can see a row of all zeros has been produced at the bottom. | For example, the [[mapping]] matrix {{rket|{{map|41 65 95 115}} {{map|31 49 72 87}} {{map|19 30 44 53}}}} is full-rank. We can check this by putting it into [[Hermite normal form]] (HNF), {{rket|{{map|1 0 0 -5}} {{map|0 1 0 2}} {{map|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, {{rket|{{map|41 65 95 115}} {{map|31 49 72 87}} {{map|10 16 23 28}}}} is rank-deficient, because its HNF is {{rket|{{map|1 1 3 3}} {{map|0 6 -7 -2}} {{map|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 <code>MatrixRank[]</code>, 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. | In Wolfram Language, an even quicker check for full-rank is possible, using <code>MatrixRank[]</code>, which will give you the count of linearly independent rows of a matrix. If this is less than both the count of rows and the count of columns, the matrix is rank-deficient. | ||
You can guarantee a full-rank result by putting a matrix into [[canonical form]]. | You can guarantee a full-rank result by putting a matrix into [[canonical form]]. | ||
== See also == | == See also == | ||
Revision as of 20:01, 29 December 2022
A matrix is full-rank when either all of its rows are linearly independent or all of its columns 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 both the count of rows and the count of columns, the matrix is rank-deficient.
You can guarantee a full-rank result by putting a matrix into canonical form.