Full-rank: Difference between revisions

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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.

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]].

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 ==

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-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.
-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]].
-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 ==