Xenharmonic Wiki

Linear dependence: Difference between revisions

← Older edit
VisualWikitext
Cmloegcmluin (talk | contribs)
autopatrolled, emailconfirmed
5,170 edits
more information about full-rank and rank-deficiency
m +Wikipedia box
 
(3 intermediate revisions by one other user not shown)
Line 1: Line 1:
[[Basis|Bases]], such as [[Comma basis|comma bases]], are considered '''linearly dependent''' when they share a common basis [[Wikipedia:Vector|vector]]. In other words, that they can form an identical vector through [[Wikipedia:Linear_combinations|linear combinations]] of their member vectors.  
[[Basis|Bases]], such as [[Comma basis|comma bases]], are considered '''linearly dependent''' when they share a common basis [[Wikipedia:Vector|vector]]. In other words, that they can form an identical vector through [[Wikipedia:Linear_combinations|linear combinations]] of their member vectors.  
{{Wikipedia|Linear independence}}


When basis vector sets do not share a common basis vector like this, they are '''linearly ''in''dependent'''. Linearly dependent basis vector sets are in a sense more closely related to each other than linearly independent basis vector sets.
When basis vector sets do not share a common basis vector like this, they are '''linearly ''in''dependent'''. Linearly dependent basis vector sets are in a sense more closely related to each other than linearly independent basis vector sets.
Line 53: Line 55:
=== Wedge product ===
=== Wedge product ===


Linear dependence has an interesting effect on the wedge product, which otherwise produces the same result on a set of vectors as one would get by treating those same vectors as basis matrices and performing a [[temperament merging|temperament merge]]. The wedge product of any two linear dependent multivectors will have all zeros for entries, and thereby not represent an interesting new temperament (whereas the wedge product for linearly independent multivectors ''does'' represent an interesting new temperament sharing properties of the input temperaments) (and where the equivalent temperament merge operation in linear algebra would provide such an interesting temperament). For more information, see: [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#Linearly dependent exception]]
Linear dependence has an interesting effect on the wedge product. Ordinarily, the wedge product can be used with multivectors to find new temperaments in an equivalent way to how new temperaments are found using concatenation with matrices. However, if the wedged multivectors are linearly dependent, then the multivector resulting from their wedge product will have all zeros for entries, and therefore it will not represent any temperament. Linear dependence does not impose this limitation on the concatenation of matrices approach, however; if equivalent linearly dependent matrices are concatenated, then a new matrix representing a new temperament will be produced. For more information, see [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#The linearly dependent exception to the wedge product]].


=== Temperament addition ===
=== Temperament addition ===
Line 70: Line 72:


==Rank-deficiency and full-rank==
==Rank-deficiency and full-rank==
 
[[File:Full-rank - updated.png|thumb|600x600px|Rank of a matrix can be revealed by reducing a matrix such as with Hermite normal form (row or column style). The shapes of the reduced matrices are shown in green within the original matrix, and the all-zero rows and columns in red. ]]
A matrix is '''[[full-rank]]''' when its [[Wikipedia:Rank (linear algebra)|rank]] equals whichever is smaller between its width (column count) and height (row count):
A matrix is '''[[full-rank]]''' when its [[Wikipedia:Rank (linear algebra)|rank]] equals whichever is smaller between its width (column count) and height (row count):
* For a ''wide'' matrix (height is smaller), it is full-rank when its rank equals its ''height''.
* For a ''wide'' matrix (height is smaller), it is full-rank when its rank equals its ''height''.
Line 107: Line 109:


===Why row-rank always equals column-rank===
===Why row-rank always equals column-rank===
 
[[File:Row-rank is col-rank.png|left|thumb|600x600px|Here we see an example matrix 𝐴 expressed as the product of matrices 𝑋 and 𝑌. These two matrices' shared dimension, 𝑟, is both the row-rank and the column-rank of 𝐴. In the middle and bottom rows of this diagram, we see how 𝐴 can be described in two different ways: in the middle row we see each of its rows as a linear combination of the 𝑟 rows of 𝑌, and in the bottom row we see each of its columns as a linear combination of the 𝑟 columns of 𝑋. So whether we reduce it row-style (HNF) or column-style (CHNF), we find the rank to be 𝑟. ]]
Any <math>(m,n)</math>-shaped matrix <math>A</math> can be expressed as the product of a <math>(m,r)</math>-shaped matrix <math>X</math> and a <math>(r,n)</math>-shaped matrix <math>Y</math>, such that the <math>r</math>-dimensions cancel out in the middle and we're left with an <math>(m,n)</math>-shaped matrix.  
Any <math>(m,n)</math>-shaped matrix <math>A</math> can be expressed as the product of a <math>(m,r)</math>-shaped matrix <math>X</math> and a <math>(r,n)</math>-shaped matrix <math>Y</math>, such that the <math>r</math>-dimensions cancel out in the middle and we're left with an <math>(m,n)</math>-shaped matrix.  


Line 121: Line 123:


  <nowiki>
  <nowiki>
reverseEachRow[a_] := Reverse[a, 2];
reverseEachCol[a_] := Reverse[a];
antitranspose[a_] := reverseEachRow[reverseEachCol[a]];
P = {
P = {
   {1,  4/3,  4/3},
   {   1,  4/3,  4/3 },
   {0, -4/3, -1/3},
   {   0, -4/3, -1/3 },
   {0,  4/3,  1/3}
   {   0,  4/3,  1/3 }
};
};
Last[HermiteDecomposition[P]]
Last[HermiteDecomposition[P]]
antitranspose[Last[HermiteDecomposition[antitranspose[P]]]]  
Last[HermiteDecomposition[Transpose[P]]


→ {
→ {
     { 1,   0,   1},
     {   1,   0,   1 },
     { 0, 4/3, 1/3},
     {   0, 4/3, 1/3 },
     { 0,   0,   0}
     {   0,   0,   0 }
   }  
   }  
→ {
→ {
     { 0,   0,   0},
     { 1/3, 2/3, -2/3 },
     {-1,  4,  0},
     {   0,    1,  -1 },
     { 0, 4/3, 1/3}
     {   0,   0,   0 }
   } </nowiki>
   } </nowiki>


On the other hand, we have the [[minimax-E-copfr-S]] (or primes miniRMS-U) tuning of 12-ET, where <math>d</math> still equals <math>3</math> but now <math>r = 1</math>:
On the other hand, we have the [[minimax-E-copfr-S]] (or primes miniRMS-U) tuning of 12-ET, where <math>d</math> still equals <math>3</math> but now <math>r = 1</math>:


  <nowiki>M = {{12, 19, 28}};
  <nowiki>M = {{ 12, 19, 28 }};
G = PseudoInverse[M]
G = PseudoInverse[M]
→ {{12, 19, 28}} / 1289
→ {{ 12, 19, 28 }} / 1289


P = G.M
P = G.M
→ {
→ {
     { 12² , 12·19, 12·28},
     { 12² , 12·19, 12·28 },
     {19·12,  19² , 19·28},
     { 19·12,  19² , 19·28 },
     {28·12, 28·19,  28² }
     { 28·12, 28·19,  28² }
   } / 1289
   } / 1289


Line 160: Line 158:
antitranspose[Last[HermiteDecomposition[antitranspose[P]]]]  
antitranspose[Last[HermiteDecomposition[antitranspose[P]]]]  
→ {
→ {
     {12, 19, 28},
     { 12, 19, 28 },
     { 0,  0,  0},
     { 0,  0,  0 },
     { 0,  0,  0}
     { 0,  0,  0 }
   }/1289
   }/1289
→ {
→ {
     { 0, 0, 0},
     { 12, 19, 28 },
     { 0,  0,  0},
     { 0,  0,  0 },
     {12, 19, 28}
     { 0, 0, 0 }
   }/1289 </nowiki>
   }/1289 </nowiki>


Retrieved from "https://en.xen.wiki/w/Linear_dependence"