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Make det, transpose, trace and an entry nodes that apply to a matrix argument - #2436
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…argument det(A) of a symbol was refused at parsing, transpose(x) too, trace(A) was the product trace * A, and A[i, j] did not parse. Each of the four is now a node over what it is applied to, carried out once that is a matrix: by Substitute, Simplify, Evaled and a compiled delegate taking T[,]. Of a matrix written out each is still computed as it is parsed, as det and transpose were, so nothing #2180 answered moves but transpose(x). - Determinantf, Transposef, Tracef and MatrixEntryf, with printing that parses back, LaTeX (\det, ^{\mathrm{T}}, \operatorname{tr}, a subscript), SymPy, substitution, sorting, serialization, and MathS.Matrices.Determinant, Transpose, Trace and Entry. - An entry counts its row and column from 1, as a_ij does: [[1, 2], [3, 4]][2, 1] is 3. The brackets follow an atom directly, which did not parse before. - A non-square matrix has no determinant and no trace, and an index out of range or not whole names no entry: the node is left as it is, never a number. Written out, each is refused at parsing, as det of a non-square matrix was. A number, a statement or a set is no matrix and is refused. - det(transpose(A)) is det(A), trace(transpose(A)) is trace(A), transpose(transpose(A)) is A and transpose(A)[k, l] is A[l, k]. - The transpose, the trace and an entry are differentiated through the matrix; the determinant keeps its derivative node, Jacobi's formula needing the adjugate. - Compiled: the transpose, Bareiss' fraction-free determinant with the largest pivot where the elements have a size (exact over long and int), the trace, and the entry with its indices checked whole and in range when called. Shapes written into the expression are checked at Compile. - A product with a transpose in it is a product of matrices, and Simplify read it as commutative: transpose(A) * A - A * transpose(A) simplified to 0, and transpose(A) * B printed as B * transpose(A). Such a product is set aside as a name of its own while the rest is simplified, and put back with each factor simplified in place; the canonical sort and the print order leave it as written. - A name beside a written matrix may stand for a matrix, and simplification reads it as a number: det(A [[2, 0], [0, 3]]) was 6 A^2 at parsing, and a compiled delegate computed that. Such an argument is not carried out, at parsing, in evaluation or by Simplify, until the name is a matrix; det([[x, 1], [2, x]]), whose names are entries, is x^2 - 2 as before. Measured: the new MatrixOperationNodesTest compares every value with one computed from the entries, at real, negative and non-real matrices, 2x2 and 3x3, and every compiled value with evaluation, including a matrix argument beside a written matrix, which the base commit's SimplifiedAroundMatrixArguments leaves to these nodes. The 2.6.0 values in BREAKING-CHANGES.md are measured on the 2.6.0 package. Closes #2427. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
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Closes #2427.
det,transpose,traceand an entryA[k, l]are now nodes. They can be written of a name, such asdet(A),transpose(A) * B,trace(A * B)orA[1, 2], and are carried out once the name is a matrix: by substitution, simplification, evaluation, and compilation overT[,](#2429).Parsing:
det,transposeandtraceof a name were read as products on 2.6.0, sodet(A)wasdet * A.det([[1, 2], [3, 4]])is-2.traceandM[k, l]are new. A bracket straight after an atom did not parse before, so that syntax adds rather than changes.det(5)throws.Indices count from 1, as
a_ijdoes, and as in Mathematica, Maple, Maxima, MATLAB and Julia. The C#Matrixindexer still counts from 0, as C# does, soM[k, l]in an expression ismatrix[k - 1, l - 1]in code. The syntax page says so.No value is invented: a non-square matrix has no determinant or trace, and an index out of range or not whole names no entry. Each stays unevaluated.
Rewrites:
transpose(transpose(A)) → A,det(transpose(A)) → det(A),trace(transpose(A)) → trace(A)andtranspose(A)[k, l] → A[l, k].Derivatives are taken through the matrix for transpose, trace and an entry. The determinant keeps its
derivativenode, since Jacobi's formula needs an adjugate.Compiled:
long;Known shapes are checked at
Compile.API:
MathS.Matrices.Determinant,Transpose,TraceandEntry. They are recorded inPublicApi.txtwith the four node records and the compiled helpers.Two wrong answers that these nodes would have made easy to reach are fixed. Both come from a name having no type yet (#1237):
transpose(A) * A - A * transpose(A)simplified to0, andtranspose(A) * BtoB * transpose(A).transpose(A)*B + transpose(A)*Cis no longer factored.A*B - B*Astill gives0, since nothing yet saysAis a matrix. That waits for Typed variables #1237.det(A * [[2, 0], [0, 3]])came out as6 * A^2.det([[x, 1], [2, x]])is stillx^2 - 2."det(A)"det * Adet(A), aDeterminantf"det(A)"withA = [[1, 2], [3, 4]], evaluated[[det, det * 2], [det * 3, det * 4]]-2"transpose(A) * B"transpose * A * Btranspose(A) * B"trace([[1, 2], [3, 4]])"trace * [[1, 2], [3, 4]]1 + 4"A[1, 2]","[[1, 2], [3, 4]][2, 1]"UnhandledParseExceptionA[1, 2];3"det(5)"det * 5InvalidArgumentParseException"det(x * [[1, 2], [3, 4]])"det * x * [[1, 2], [3, 4]]xmay stand for a matrixThe rows are in BREAKING-CHANGES.md, measured on the 2.6.0 package. #2180's unshipped
transpose(x)row, which no longer throws, is updated. The parser was regenerated with the vendored ANTLR; regenerating the unmodified grammar first gave an empty diff.Checked on this branch:
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