Linear dependence: Difference between revisions
Cmloegcmluin (talk | contribs) →Rank-deficiency and full-rank: add diagrams |
Cmloegcmluin (talk | contribs) →Why row-rank always equals column-rank: eliminate unnecessary use of antitranspose; make same point more simply with a single transpose |
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<nowiki> | <nowiki> | ||
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]] | ||
Last[HermiteDecomposition[Transpose[P]] | |||
→ { | → { | ||
{ 1, | { 1, 0, 1 }, | ||
{ 0, 4/3, 1/3}, | { 0, 4/3, 1/3 }, | ||
{ 0, | { 0, 0, 0 } | ||
} | } | ||
→ { | → { | ||
{ | { 1/3, 2/3, -2/3 }, | ||
{ | { 0, 1, -1 }, | ||
{ 0, | { 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 156: | ||
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 | ||
→ { | → { | ||
{ | { 12, 19, 28 }, | ||
{ 0, 0, 0}, | { 0, 0, 0 }, | ||
{ | { 0, 0, 0 } | ||
}/1289 </nowiki> | }/1289 </nowiki> | ||