Full-rank - Revision history

Cmloegcmluin: link to more information

2022-12-31T19:07:42Z

link to more information

← Older revision		Revision as of 19:07, 31 December 2022
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	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 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]].
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	== See also ==		== See also ==

			* [[Linear dependence#Rank-deficiency and full-rank]]: for a more in-depth textbook tutorial style look at this concept and how it relates to RTT
	* [[Wikipedia: Rank (linear algebra) #Main_definitions\|Related terminology on Wikipedia]]		* [[Wikipedia: Rank (linear algebra) #Main_definitions\|Related terminology on Wikipedia]]

Cmloegcmluin: correct article; no such thing as full-grade, full-nullity, grade-deficient, or nullity-deficient

2022-12-29T20:01:33Z

correct article; no such thing as full-grade, full-nullity, grade-deficient, or nullity-deficient

← Older revision		Revision as of 20:01, 29 December 2022
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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.		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]].

	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 ==		== See also ==

Cmloegcmluin: update EBK style

2022-12-27T17:32:10Z

update EBK style

← Older revision		Revision as of 17:32, 27 December 2022
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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 all of its rows are [[linearly independent]]. Otherwise, it is '''rank-deficient'''.

	For example, the [[mapping]] matrix {{~~ket~~\|{{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), {{~~ket~~\|{{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, {{~~ket~~\|{{map\|41 65 95 115}} {{map\|31 49 72 87}} {{map\|10 16 23 28}}}} is rank-deficient, because its HNF is {{~~ket~~\|{{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 the count of rows, the matrix is rank-deficient.

Cmloegcmluin at 19:19, 30 December 2021

2021-12-30T19:19:03Z

← Older revision		Revision as of 19:19, 30 December 2021
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	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]].

	One could generalize this notion to full-nullity and nullity-deficient when speaking of the linear independence of columns of a [[comma basis]].		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 ==		== See also ==

Cmloegcmluin: consolidate categories

2021-12-25T13:38:06Z

consolidate categories

← Older revision		Revision as of 13:38, 25 December 2021
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	* [[Wikipedia: Rank (linear algebra) #Main_definitions\|Related terminology on Wikipedia]]		* [[Wikipedia: Rank (linear algebra) #Main_definitions\|Related terminology on Wikipedia]]

	~~[[Category:Temperament]]~~
	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	~~[[Category:Theory]]~~
	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Math]]		[[Category:Math]]

Cmloegcmluin at 16:17, 20 December 2021

2021-12-20T16:17:07Z

← Older revision		Revision as of 16:17, 20 December 2021
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	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 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]].		One could generalize this notion to full-nullity and nullity-deficient when speaking of the linear independence of columns of a [[comma basis]].

Cmloegcmluin: see also, and categories

2021-12-11T18:42:01Z

Cmloegcmluin: Created page with "A matrix is '''full-rank''' when all of its rows are linearly independent. Otherwise, it is '''rank-deficient'''. For example, the mapping matrix {{ket|{{map|41 65 95..."

2021-12-11T18:38:15Z

Created page with "A matrix is '''full-rank''' when all of its rows are linearly independent. Otherwise, it is '''rank-deficient'''. For example, the mapping matrix {{ket|{{map|41 65 95..."

New page

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

For example, the [[mapping]] matrix {{ket|{{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), {{ket|{{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, {{ket|{{map|41 65 95 115}} {{map|31 49 72 87}} {{map|10 16 23 28}}}} is rank-deficient, because its HNF is {{ket|{{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.

One could generalize this notion to full-nullity and nullity-deficient when speaking of the linear independence of columns of a [[comma basis]].

← Older revision		Revision as of 18:42, 11 December 2021
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	One could generalize this notion to full-nullity and nullity-deficient when speaking of the linear independence of columns of a [[comma basis]].		One could generalize this notion to full-nullity and nullity-deficient when speaking of the linear independence of columns of a [[comma basis]].

			== See also ==

			* [[Wikipedia: Rank (linear algebra) #Main_definitions\|Related terminology on Wikipedia]]

			[[Category:Temperament]]
			[[Category:Regular temperament theory]]
			[[Category:Theory]]
			[[Category:Terms]]
			[[Category:Math]]