Array programming: Difference between revisions

Revision as of 18:41, 28 June 2021

Revision as of 15:58, 25 July 2021

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Both MATLAB and GNU Octave natively support [[linear algebra]] operations such as matrix multiplication, [[matrix inversion]], and the numerical solution of [[system of linear equations]], even using the [[Moore–Penrose pseudoinverse]].<ref>{{cite web |title= GNU Octave Manual. Arithmetic Operators. |url= https://www.gnu.org/software/octave/doc/interpreter/Arithmetic-Ops.html |access-date= 2011-03-19}}</ref><ref>{{cite web |title= MATLAB documentation. Arithmetic Operators. |url= http://www.mathworks.com/help/techdoc/ref/arithmeticoperators.html |access-date= 2011-03-19}}</ref>

The [[Nial]] example of the inner product of two arrays can be implemented using the native matrix multiplication operator. If <code>a</code> is a row vector of size [1 n] and <code>b</code> is a corresponding column vector of size [n 1].

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==Mathematical reasoning and language notation==

The matrix left-division operator concisely expresses some semantic properties of matrices. As in the scalar equivalent, if the ([[determinant]] of the) coefficient (matrix) <code>A</code> is not null then it is possible to solve the (vectorial) equation <code>A * x = b</code> by left-multiplying both sides by the [[matrixinverse inversionmatrix|inverse]] of <code>A</code>: <code>A⁻¹</code> (in both MATLAB and GNU Octave languages: <code>A^-1</code>). The following mathematical statements hold when <code>A</code> is a [[matrix rank#Properties|full rank]] [[square matrix#Square matrices|square matrix]]:

:<code>A^-1 *(A * x)==A^-1 * (b)</code>

:<code>(A^-1 * A)* x ==A^-1 * b </code> &nbsp; &nbsp; &nbsp; (matrix-multiplication [[associativity]])

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The previous statements are also valid MATLAB expressions if the third one is executed before the others (numerical comparisons may be false because of round-off errors).

If the system is overdetermined -– so that <code>A</code> has more rows than columns -– the pseudoinverse <code>A⁺</code> (in MATLAB and GNU Octave languages: <code>pinv(A)</code>) can replace the inverse <code>A⁻¹</code>, as follows:

:<code>pinv(A) *(A * x)==pinv(A) * (b)</code>

:<code>(pinv(A) * A)* x ==pinv(A) * b</code> &nbsp; &nbsp; &nbsp; (matrix-multiplication associativity)