Matrix Mapping and Equation Solving

Tools for converting GA multivector products into matrices and solving the resulting linear systems. Part of the Tan.GA matrix pipeline; see Product matrices for the conceptual overview.

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Involution Flags

All product-matrix functions in Matrix_Product.h accept optional eInvLeft and eInvRight parameters (type GA::EInv, default GA::EInv::Id):

Value	Effect
GA::EInv::Id	Identity — no sign change
GA::EInv::Rev	Reverse: rev(blade) = (-1)^(k(k-1)/2) * blade
GA::EInv::Conj	Clifford conjugate: conj(blade) = rev(blade) * (-1)^r

The involution sign is applied per-blade:

• eInvLeft modifies the A-blade coefficient before the inner loop: fValA_signed = rev_sign(blA) ? -fValA : fValA
• eInvRight modifies the B-blade column sign inside the inner loop: finalSign = productSign ⊕ rev_sign(blB)

This enables building product matrices for equations such as rev(A) * X = C, A * rev(X) = C, or conj(A) * conj(X) = C without pre-computing the involuted operands.

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Reverse and Conjugate Sign Matrices

EvalProductMatrix_Reverse and EvalProductMatrix_Conjugate build diagonal |xMask| × |xMask| matrices where M[i,i] = ±1 for each blade in xMask. These are involutions: M² = I.

GA::EvalProductMatrix_Reverse(mat, xMask);    // Reverse sign on diagonal
GA::EvalProductMatrix_Conjugate(mat, xMask);  // Conjugate sign on diagonal

Use with vec(rev(A)) = D_rev · vec(A) when constructing systems that involve involution operations as part of the unknown.

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Complete API Reference

Blade Mask Evaluation

// Collect blade ids from a multivector
GA::EvalBladeMask(xMask, wA, bOnlyNonZeroComps);

Blade Mask Prediction

// Predict output blade mask of A ∘ X
GA::EvalProductBladeMask_GP(xMaskC, wA, xMaskB, bLeftToRight, bComplete);
GA::EvalProductBladeMask_IP(xMaskC, wA, xMaskB, bLeftToRight, bComplete);
GA::EvalProductBladeMask_OP(xMaskC, wA, xMaskB, bLeftToRight, bComplete);

// Mask-based overloads (use xMaskA instead of wA)
GA::EvalProductBladeMask_GP(xMaskC, xMaskA, xMaskB, bLeftToRight, bComplete);
GA::EvalProductBladeMask_IP(xMaskC, xMaskA, xMaskB, bLeftToRight, bComplete);
GA::EvalProductBladeMask_OP(xMaskC, xMaskA, xMaskB, bLeftToRight, bComplete);

// Inverse prediction: given A and C, what can X be?
GA::EvalProductBladeMaskInv_GP(xMaskB, xMaskA, xMaskC, bLeftToRight);
GA::EvalProductBladeMaskInv_IP(xMaskB, xMaskA, xMaskC, bLeftToRight);
GA::EvalProductBladeMaskInv_OP(xMaskB, xMaskA, xMaskC, bLeftToRight);

MV ↔ Matrix Conversion

GA::ToMatrix(mat, wA, xMask);           // MV to column matrix
GA::ToMultivector(wA, mat, xMask);      // column matrix to MV

Product Matrix Construction

// 2-mask overloads
GA::EvalProductMatrix_GP(mat, wA, xMaskB, xMaskC, bLeftToRight,
                         eInvLeft, eInvRight);
GA::EvalProductMatrix_IP(mat, wA, xMaskB, xMaskC, bLeftToRight,
                         eInvLeft, eInvRight);
GA::EvalProductMatrix_OP(mat, wA, xMaskB, xMaskC, bLeftToRight,
                         eInvLeft, eInvRight);

// 3-mask overloads (restrict A to xMaskA)
GA::EvalProductMatrix_GP(mat, wA, xMaskA, xMaskB, xMaskC, bLeftToRight,
                         eInvLeft, eInvRight);
GA::EvalProductMatrix_IP(mat, wA, xMaskA, xMaskB, xMaskC, bLeftToRight,
                         eInvLeft, eInvRight);
GA::EvalProductMatrix_OP(mat, wA, xMaskA, xMaskB, xMaskC, bLeftToRight,
                         eInvLeft, eInvRight);

// Array overloads (stacked matrix from list of MVs)
GA::EvalProductMatrixArray_GP(mat, vecwListA, xMaskB, xMaskC, bLeftToRight,
                               eInvLeft, eInvRight);
GA::EvalProductMatrixArray_IP(mat, vecwListA, xMaskB, xMaskC, bLeftToRight,
                               eInvLeft, eInvRight);
GA::EvalProductMatrixArray_OP(mat, vecwListA, xMaskB, xMaskC, bLeftToRight,
                               eInvLeft, eInvRight);

// Reverse / Conjugate sign matrices (diagonal, from blade mask only)
GA::EvalProductMatrix_Reverse(mat, xMask);
GA::EvalProductMatrix_Conjugate(mat, xMask);

All EInv parameters default to GA::EInv::Id for backward compatibility.

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Related Files

File	Purpose
Tan.GA/Matrix_BladeMask.h	Blade mask prediction
Tan.GA/Matrix_Product.h	Product matrix construction
Tan.GA/Matrix_MapToBladeMask.h	Umbrella header
Tan.GA/Enum.h	EInv enum
Tan.Math/Matrix.Algo.GE.h	Gaussian elimination
Tan.Math/Matrix.Algo.SVD.h	SVD solver
Tan.Math/Congruence.h	Modular congruence classes