Matrix Inverse/Divide: Difference between revisions

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'''Caution:''' Be careful not to confuse this symbol, which is <apll>⌹</apll>, with <apll>⍠</apll> which is [[Variant]].
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   <td><apll>R</apll> is a numeric scalar, vector or matrix; otherwise signal a <apll>RANK ERROR</apll>.  If <apll>R</apll> is a matrix, <apll>≥/⍴R</apll> must be true; otherwise signal a <apll>LENGTH ERROR</apll>.</td>
   <td><apll>R</apll> is a numeric scalar, vector or matrix; otherwise signal a <apll>RANK ERROR</apll>.</td>
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<p>This feature implements matrix inversion on integer or floating point arguments using [http://www.gnu.org/software/gsl/manual/html_node/Singular-Value-Decomposition.html Singular Value Decomposition].  In particular, this means that any numeric array meeting the rank and shape requirements above is invertible.</p>
<p>This feature implements matrix inversion on '''Boolean''', '''integer''', or '''floating point''' arguments using [http://www.gnu.org/software/gsl/manual/html_node/Singular-Value-Decomposition.html Singular Value Decomposition].  In particular, this means that any numeric array meeting the rank and shape requirements above is invertible.</p>


<p>Matrix inverse (<apll>⌹R</apll>) and matrix division (<apll>L⌹R</apll>) on Rational or VFP arguments each have two limitations above and beyond that of normal conformability:</p>
<p>Matrix inverse (<apll>⌹R</apll>) and matrix division (<apll>L⌹R</apll>) on '''Rational''', '''VFP''', or '''Ball''' arguments each have two limitations above and beyond that of normal conformability:</p>
     <ul>
     <ul>
       <li><p>for a square right argument that it be non-singular, and</p></li>
       <li><p>for a '''square''' (<apll>=/⍴R</apll>) right argument that it be non-singular, and</p></li>
       <li><p>for an overdetermined (<apll>&gt;/⍴R</apll>) right argument that the symmetric matrix <apll>(⍉R)+.×R</apll> be non-singular.</p></li>
       <li><p>for an '''overdetermined''' (<apll>&gt;/⍴R</apll>) or '''underdetermined''' (<apll>&lt;/⍴R</apll>) right argument that the symmetric matrix <apll>(⍉R)+.×R</apll> be non-singular.</p></li>
     </ul>
     </ul>
     <p>These limitations are due to the algorithm ([http://en.wikipedia.org/wiki/Gauss%E2%80%93Jordan_elimination Gauss-Jordan Elimination]) used to implement Matrix Inverse/Divide on Rational and VFP numbers.</p>
     <p>These limitations are due to the algorithm ([http://en.wikipedia.org/wiki/Gauss%E2%80%93Jordan_elimination Gauss-Jordan Elimination]) used to implement Matrix Inverse/Divide on Rational and VFP numbers.</p>
<p>Overdetermined matrices are evaluated equivalently to the expression <apll>(⌹(⍉R)+.×R)+.×⍉R</apll>.</p>
<p>Underdetermined matrices are evaluated equivalently to the expression <apll>(⍉R)+.×⌹R+.×⍉R</apll>.</p>


<p>For example,</p>
<p>For example,</p>


<apll>
<apll><pre>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⌹3 3{rho}0<br />
      ⌹3 3{rho}0
&nbsp;0 0 0<br />
0 0 0
&nbsp;0 0 0<br />
0 0 0
&nbsp;0 0 0<br />
0 0 0
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⌹3 3⍴1 2 3 4<br />
      ⌹3 3⍴1 2 3 4
¯0.1944444444&nbsp;&nbsp;&nbsp;0.2777777778&nbsp;&nbsp;&nbsp;0.02777777778<br />
¯0.1944444444   0.2777777778   0.02777777778
&nbsp;0.05555555556&nbsp;¯0.2222222222&nbsp;&nbsp;&nbsp;0.2777777778<br />
0.05555555556 ¯0.2222222222   0.2777777778
&nbsp;0.3611111111&nbsp;&nbsp;&nbsp;0.05555555556&nbsp;¯0.1944444444<br />
0.3611111111   0.05555555556 ¯0.1944444444
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⌹3 3⍴1 2 3 4x<br />
      ⌹3 3⍴1 2 3 4<pn>x</pn>
¯7r36&nbsp;&nbsp;5r18&nbsp;&nbsp;1r36<br />
¯7<pn>r</pn>36  5<pn>r</pn>18  1<pn>r</pn>36
&nbsp;1r18&nbsp;¯2r9&nbsp;&nbsp;&nbsp;5r18<br />
1<pn>r</pn>18 ¯2<pn>r</pn>9  5<pn>r</pn>18
13r36&nbsp;&nbsp;1r18&nbsp;¯7r36<br />
13<pn>r</pn>36  1<pn>r</pn>18 ¯7<pn>r</pn>36
</apll>
</pre></apll>
<br />
 
<br />
<h2>Variations On Matrix Divide</h2>
 
<p>For variations on this function, see the [[Variant#Eigenvalues_and_Eigenvectors|Variant Operator]].</p>
 
 
 
 
 
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<p>This feature implements matrix division on integers and floating point arguments using [http://www.gnu.org/software/gsl/manual/html_node/Singular-Value-Decomposition.html Singular Value Decomposition].  In particular, this means that any two numeric arrays meeting the rank and shape requirements above are divisible.</p>
<p>This feature implements matrix division on '''Boolean''', '''integer''' and '''floating point''' arguments using [http://www.gnu.org/software/gsl/manual/html_node/Singular-Value-Decomposition.html Singular Value Decomposition].  In particular, this means that any two numeric arrays meeting the rank and shape requirements above are divisible.</p>
 
<p>As noted above, Matrix Division on '''Rational''', '''VFP''', or '''Ball''' arguments uses a different algorithm and has slightly more restrictive conformability requirements.</p>
    <ul>
      <li><p>for a '''square''' (<apll>=/⍴R</apll>) right argument that it be non-singular, and</p></li>
      <li><p>for an '''overdetermined''' (<apll>&gt;/⍴R</apll>) or '''underdetermined''' (<apll>&lt;/⍴R</apll>) right argument that the symmetric matrix <apll>(⍉R)+.×R</apll> be non-singular.</p></li>
    </ul>


<p>As noted above, Matrix Division on Rational or VFP arguments uses a different algorithm and has slightly more restrictive conformability requirements.</p>
<p>Overdetermined matrices are evaluated equivalently to the expression <apll>(⌹(⍉R)+.×R)+.×(⍉R)+.×L</apll>.</p>
<p>Underdetermined matrices are evaluated equivalently to the expression <apll>(⍉R)+.×⌹R+.×(⍉R)+.×L</apll>.</p>


<p>For example,</p>
<p>For example,</p>


<apll>
<apll><pre>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;a←3 3⍴0<br />
      a←3 3⍴0
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;a⌹a<br />
      a⌹a
&nbsp;0 0 0<br />
0 0 0
&nbsp;0 0 0<br />
0 0 0
&nbsp;0 0 0<br />
0 0 0
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1 2 3⌹3 3⍴1 2 3 4<br />
      1 2 3⌹3 3⍴1 2 3 4
0.4444444444 0.4444444444 ¯0.1111111111<br />
0.4444444444 0.4444444444 ¯0.1111111111
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1 2 3⌹3 3⍴1 2 3 4x<br />
      1 2 3⌹3 3⍴1 2 3 4<pn>x</pn>
4r9 4r9 ¯1r9</apll>
4<pn>r</pn>9 4<pn>r</pn>9 ¯1<pn>r</pn>9</pre></apll>
 
<hr width=35%><br>
 
<big>For several more everyday problem-solving examples using Matrix Inverse see also '''[[Symbol_Domino|Domino Symbol ⌹]]'''.</big>


<br><hr width=35%>


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Latest revision as of 09:58, 8 October 2023

Caution: Be careful not to confuse this symbol, which is ⌹, with ⍠ which is Variant.

Z←⌹R returns the inverse of the right argument.
R is a numeric scalar, vector or matrix; otherwise signal a RANK ERROR.
Z is a numeric array of rank ⍴⍴R, and shape ⌽⍴R.


This feature implements matrix inversion on Boolean, integer, or floating point arguments using Singular Value Decomposition. In particular, this means that any numeric array meeting the rank and shape requirements above is invertible.

Matrix inverse (⌹R) and matrix division (L⌹R) on Rational, VFP, or Ball arguments each have two limitations above and beyond that of normal conformability:

  • for a square (=/⍴R) right argument that it be non-singular, and

  • for an overdetermined (>/⍴R) or underdetermined (</⍴R) right argument that the symmetric matrix (⍉R)+.×R be non-singular.

These limitations are due to the algorithm (Gauss-Jordan Elimination) used to implement Matrix Inverse/Divide on Rational and VFP numbers.

Overdetermined matrices are evaluated equivalently to the expression (⌹(⍉R)+.×R)+.×⍉R.

Underdetermined matrices are evaluated equivalently to the expression (⍉R)+.×⌹R+.×⍉R.

For example,

      ⌹3 3⍴0
 0 0 0
 0 0 0
 0 0 0
      ⌹3 3⍴1 2 3 4
¯0.1944444444   0.2777777778   0.02777777778
 0.05555555556 ¯0.2222222222   0.2777777778
 0.3611111111   0.05555555556 ¯0.1944444444
      ⌹3 3⍴1 2 3 4x
¯7r36  5r18  1r36
 1r18 ¯2r9   5r18
13r36  1r18 ¯7r36

Variations On Matrix Divide

For variations on this function, see the Variant Operator.



Z←L⌹R returns the quotient of the left and right arguments. This quotient can be interpreted in various ways, such as the least squares solution of the system of linear equations determined by arguments.
L is a numeric scalar, vector or matrix; otherwise signal a RANK ERROR.
R is a numeric scalar, vector or matrix; otherwise signal a RANK ERROR.
If either L or R is a scalar or vector, it is coerced to a matrix by (L R)←⍪¨L R. After this coercion, if the two matrices have a different number of rows, signal a LENGTH ERROR.
Z is a numeric array. Before the above coercion of L and R, the rank of Z is ¯2+2⌈(⍴⍴L)+⍴⍴R, and the shape is (1↓⍴R),1↓⍴L.


This feature implements matrix division on Boolean, integer and floating point arguments using Singular Value Decomposition. In particular, this means that any two numeric arrays meeting the rank and shape requirements above are divisible.

As noted above, Matrix Division on Rational, VFP, or Ball arguments uses a different algorithm and has slightly more restrictive conformability requirements.

  • for a square (=/⍴R) right argument that it be non-singular, and

  • for an overdetermined (>/⍴R) or underdetermined (</⍴R) right argument that the symmetric matrix (⍉R)+.×R be non-singular.

Overdetermined matrices are evaluated equivalently to the expression (⌹(⍉R)+.×R)+.×(⍉R)+.×L.

Underdetermined matrices are evaluated equivalently to the expression (⍉R)+.×⌹R+.×(⍉R)+.×L.

For example,

      a←3 3⍴0
      a⌹a
 0 0 0
 0 0 0
 0 0 0
      1 2 3⌹3 3⍴1 2 3 4
0.4444444444 0.4444444444 ¯0.1111111111
      1 2 3⌹3 3⍴1 2 3 4x
4r9 4r9 ¯1r9


For several more everyday problem-solving examples using Matrix Inverse see also Domino Symbol ⌹.



See Also
System Commands System Variables and Functions Operators


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