# Arithmetic Matrix Operations

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## Matrix Multiplication.

For matrices whose matrix elements are members of a ring, the binary operations of the ring can be used to define matrix multiplication. Let (R, 0, +, ·) be a ring where R is a non-empty set, + is the additive binary operator, 0 is the additive identity and · is the multiplicative binary operator. Given an m × n matrix A = (aij) and an n × p matrix B = (bij), where aij, bijR, define the m × p matrix C = A B by C = (cij), where cij = Σnk=1 aik bkj.

The functionality of matrix routines in this collection are routines which could be implemented for matrices defined over an arbitrary ring but the implementation presented here is for real matrices declared as double A[M][N] or for dynamically defined matrices declared as double* A and for complex matrices declared as double complex A[M][N] or for dynamically defined complex matrices declared as double complex *A.

The routine for multiplying two real matrices is Multiply_Matrices( ) and the routine for multiplying two complex matrices is Multiply_CMatrices( ). The routine for multiplying a real matrix and a real column vector is Multiply_Matrix_by_Vector( ) and the routine for multiplying a real row vector by a real matrix is Multiply_Vector_by_Matrix( ). The routine for multiplying a complex matrix and a complex column vector is Multiply_CMatrix_by_CVector( ) and the routine for multiplying a complex row vector by a complex matrix is Multiply_CVector_by_CMatrix( ). There are special versions of these routines for the cases of the ubiquitous 2×2 and 3×3 matrices and for the 2-d and 3-d vectors.

Additionally there are functions for multiplying a row vector by a matrix by a column vector: For real vectors and a real matrix, the function Bilinear_Pairing() returns the real number uTAv. Given real vector spaces U and V, a pairing is a bilinear function (,):U×VR. The definition of a pairing for real vector spaces can be extended to that for complex vector spaces in several ways: Given complex vector spaces U and V, a pairing is a bilinear function (,):U×VC or a pairing is a sesquilinear function (,):U×VC. In the case of the sesquilinear definition, there is a choice to which of the two arguments one applies the complex conjugation. For complex vectors and a complex matrix, the function CBilinear_Pairing() returns the complex number uTAv and the function Sesquilinear_Pairing() returns the complex number uAv, where † represents the transpose of the complex conjugate of the complex vector u, i.e. the first argument of the sesquilinear pairing is conjugated.

Given a real vector space V, a bilinear form is a bilinear function (,):V×VR. For real vectors and a real matrix, the function Bilinear_Form() returns the real number uTAv. Given a complex vector space V, a bilinear form is a bilinear function (,):V×VC or a sesquilinear form is a sesquilinear function (,):V×VC. In the case of the sesquilinear definition, there is a choice to which of the two arguments one applies the complex conjugation. For complex vectors and a complex matrix, the function CBilinear_Form() returns the complex number uTAv and the function Sesquilinear_Form() returns the complex number uAv, where † represents the transpose of the complex conjugate of the complex vector u, i.e. the first argument of the sesquilinear form is conjugated.

Special versions for multiplying a matrix by a vector for the case in which the matrix is either symmetric, skew-symmetric, lower-triangular, or Hermitian and stored in lower triangular form can be found at More Matrix Multiplication For Matrices Stored in Lower Triangular Form.

Similarly special versions for multiplying a matrix by a vector for the case in which the matrix is either symmetric, skew-symmetric, lower-triangular, or Hermitian and stored in upper triangular form can be found at More Matrix Multiplication For Matrices Stored in Upper Triangular Form.

Additional routines for multiplying complex matrices in which at least one of the summands is a real matrix can be found at More Complex Matrix Multiplication.

### Function List

• void Multiply_Matrices( double *C, double *A, int nrows, int ncols, double *B, int
mcols )

This routine calculates the matrix C = AB, where A is an nrows × ncols matrix, B is an ncols × mcols matrix and C is an nrows × mcols matrix.

• void Multiply_Matrices_2x2( double *C, double *A, double *B )

This routine calculates the matrix C = AB for the special case in which all matrices, A, B, and C, are 2×2 matrices. The matrices A, B, C should be declared in the calling routine as double .

• void Multiply_Matrices_3x3( double *C, double *A, double *B )

This routine calculates the matrix C = AB for the special case in which all matrices A, B, and C are 3×3 matrices. The matrices A, B, C should be declared in the calling routine as double .

• void Multiply_Matrix_by_Vector( double u[ ], double *A, int nrows, int ncols, double
v[ ] )

Post multiply the nrows × ncols matrix A by the column vector v to calculate the column vector u, i.e. u = A v.

• void Multiply_Vector_by_Matrix( double u[ ], double v[ ], double *A, int nrows, int
ncols )

Calculate the row vector u by pre-multiplying the nrows×ncols matrix A by the row vector v, i.e. u = v A.

• double Bilinear_Pairing(double u[], double *A, double v[], int nrows, int ncols)

Calculate uTAv, where u is nrows-dimensional column vector, v is an ncols-dimensional column vector and A is an nrows×ncols matrix.

• double Bilinear_Form(double u[], double *A, double v[], int n)

Calculate uTAv, where u is n-dimensional column vector, v is an n-dimensional column vector and A is an n×n square matrix.

• void Multiply_CMatrices( double complex *C, double complex *A, int nrows, int ncols, double complex *B, int
mcols )

This routine calculates the complex matrix C = AB, where A is an nrows × ncols complex matrix, B is an ncols × mcols complex matrix and C is an nrows × mcols matrix.

• void Multiply_CMatrices_2x2( double complex *C, double complex *A, double complex *B )

This routine calculates the complex matrix C = AB for the special case in which all matrices, A, B, and C, are 2×2 complex matrices. The matrices A, B, C should be declared in the calling routine as double complex.

• void Multiply_CMatrices_3x3( double complex *C, double complex *A, double complex *B )

This routine calculates the complex matrix C = AB for the special case in which all matrices A, B, and C are 3×3 complex matrices. The matrices A, B, C should be declared in the calling routine as double complex.

• void Multiply_CMatrix_by_CVector( double complex u[ ], double complex *A, int nrows, int ncols, double complex v[ ] )

Post multiply the nrows × ncols complex matrix A by the complex column vector v to calculate the complex column vector u, i.e. u = A v.

• void Multiply_CVector_by_CMatrix( double complex u[ ], double complex v[ ], double complex *A, int nrows, int
ncols )

Calculate the complex row vector u by pre-multiplying the nrows×ncols complex matrix A by the complex row vector v, i.e. u = v A.

• double complex CBilinear_Pairing(double complex u[], double complex *A, double complex v[], int nrows, int ncols)

Calculate uTAv, where u is nrows-dimensional complex column vector, v is an ncols-dimensional complex column vector and A is an nrows×ncols complex matrix.

• double complex CBilinear_Form(double complex u[], double complex *A, double complex v[], int n)

Calculate uTAv, where u is n-dimensional complex column vector, v is an n-dimensional complex column vector and A is an n×n complex square matrix.

• double complex Sesquilinear_Pairing(double complex u[], double complex *A, double complex v[], int nrows, int ncols)

Calculate uAv, where u is nrows-dimensional complex column vector, v is an ncols-dimensional complex column vector and A is an nrows×ncols complex matrix.

• double complex Sesquilinear_Form(double complex u[], double complex *A, double complex v[], int n)

Calculate uAv, where u is n-dimensional complex column vector, v is an n-dimensional complex column vector and A is an n×n complex square matrix.

#### C Test Code, Test Results, and Build Shell Script

• The file, testmulmatrices.c, contains a test program of Multiply_Matrices( ) in the file multiply_matrices.c. This test program requires the file multiply_matrices.c listed above.

• The file, MulMatricesTest.txt, contains the results of the test program testmulmatrices.c.

• The file, testmulmatrices.sh, contains the shell script used to compile, link, and execute the test program testmulmatrices.c.

• The file, testmulmatrices2x2.c, contains a test program of Multiply_Matrices_2x2( ) in the file multiply_matrices_2x2.c. This test program requires the file multiply_matrices_2x2.c listed above.

• The file, MulMatrices2x2Test.txt, contains the results of the test program testmulmatrices2x2.c.

• The file, testmulmatrices2x2.sh, contains the shell script used to compile, link, and execute the test program testmulmatrices2x2.c.

• The file, testmulmatrices2x2_h.c, contains a test program of Multiply_Matrices_2x2( ) in the file multiply_matrices_2x2.h. This test program requires the file multiply_matrices_2x2.h listed above.

• The file, MulMatrices2x2_hTest.txt, contains the results of the test program testmulmatrices2x2_h.c.

• The file, testmulmatrices2x2_h.sh, contains the shell script used to compile, link, and execute the test program testmulmatrices2x2_h.c.

• The file, testmulmatrices3x3.c, contains a test program of Multiply_Matrices_3x3( ) in the file multiply_matrices_3x3.c. This test program requires the file multiply_matrices_3x3.c listed above.

• The file, MulMatrices3x3Test.txt, contains the results of the test program testmulmatrices3x3.c.

• The file, testmulmatrices3x3.sh, contains the shell script used to compile, link, and execute the test program testmulmatrices3x3.c.

• The file, testmulmatrices3x3_h.c, contains a test program of Multiply_Matrices_3x3( ) in the file multiply_matrices_3x3.h. This test program requires the file multiply_matrices_3x3.h listed above.

• The file, MulMatrices3x3_hTest.txt, contains the results of the test program testmulmatrices3x3_h.c.

• The file, testmulmatrices3x3_h.sh, contains the shell script used to compile, link, and execute the test program testmulmatrices3x3_h.c.

• The file, testmulmatrixbyvector.c, contains a test program of Multiply_Matrix_by_Vector( ) in the file multiply_matrix_by_vector.c. This test program requires the file multiply_matrix_by_vector.c listed above.

• The file, MulMatrixByVectorTest.txt, contains the results of the test program testmulmatrixbyvector.c.

• The file, testmulmatrixbyvector.sh, contains the shell script used to compile, link, and execute the test program testmulmatrixbyvector.c.

• The file, testmulvectorbymatrix.c, contains a test program of Multiply_Vector_by_Matrix( ) in the file multiply_vector_by_matrix.c. This test program requires the file multiply_vector_by_matrix.c listed above.

• The file, MulVectorByMatrixTest.txt, contains the results of the test program testmulvectorbymatrix.c.

• The file, testmulvectorbymatrix.sh, contains the shell script used to compile, link, and execute the test program testmulvectorbymatrix.c.

• The file, testbilinearpairing.c, contains a test program of Bilinear_Pairing( ) in the file bilinear_pairing.c. This test program requires the file bilinear_pairing.c listed above.

• The file, BilinearPairingTest.txt, contains the results of the test program testbilinearpairing.c.

• The file, testbilinearpairing.sh, contains the shell script used to compile, link, and execute the test program testbilinearpairing.c.

• The file, testbilinearform.c, contains a test program of Bilinear_Form( ) in the file bilinear_form.c. This test program requires the file bilinear_form.c listed above.

• The file, BilinearFormTest.txt, contains the results of the test program testbilinearform.c.

• The file, testbilinearform.sh, contains the shell script used to compile, link, and execute the test program testbilinearform.c.

• The file, testmulcmatrices.c, contains a test program of Multiply_CMatrices( ) in the file multiply_cmatrices.c. This test program requires the file multiply_cmatrices.c listed above.

• The file, MulCMatricesTest.txt, contains the results of the test program testmulcmatrices.c.

• The file, testmulcmatrices.sh, contains the shell script used to compile, link, and execute the test program testmulcmatrices.c.

• The file, testmulcmatrices2x2.c, contains a test program of Multiply_CMatrices_2x2( ) in the file multiply_cmatrices_2x2.c. This test program requires the file multiply_cmatrices_2x2.c listed above.

• The file, MulCMatrices2x2Test.txt, contains the results of the test program testmulcmatrices2x2.c.

• The file, testmulcmatrices2x2.sh, contains the shell script used to compile, link, and execute the test program testmulcmatrices2x2.c.

• The file, testmulcmatrices2x2_h.c, contains a test program of Multiply_CMatrices_2x2( ) in the file multiply_cmatrices_2x2.h. This test program requires the file multiply_cmatrices_2x2.h listed above.

• The file, MulCMatrices2x2_hTest.txt, contains the results of the test program testmulcmatrices2x2_h.c.

• The file, testmulcmatrices2x2_h.sh, contains the shell script used to compile, link, and execute the test program testmulcmatrices2x2_h.c.

• The file, testmulcmatrices3x3.c, contains a test program of Multiply_CMatrices_3x3( ) in the file multiply_cmatrices_3x3.c. This test program requires the file multiply_matrices_3x3.c listed above.

• The file, MulCMatrices3x3Test.txt, contains the results of the test program testmulcmatrices3x3.c.

• The file, testmulcmatrices3x3.sh, contains the shell script used to compile, link, and execute the test program testmulcmatrices3x3.c.

• The file, testmulcmatrices3x3_h.c, contains a test program of Multiply_CMatrices_3x3( ) in the file multiply_cmatrices_3x3.h. This test program requires the file multiply_cmatrices_3x3.h listed above.

• The file, MulCMatrices3x3_hTest.txt, contains the results of the test program testmulcmatrices3x3_h.c.

• The file, testmulcmatrices3x3_h.sh, contains the shell script used to compile, link, and execute the test program testmulcmatrices3x3_h.c.

• The file, testmulcmatrixbycvector.c, contains a test program of Multiply_CMatrix_by_CVector( ) in the file multiply_cmatrix_by_cvector.c. This test program requires the file multiply_cmatrix_by_cvector.c listed above.

• The file, MulCMatrixByCVectorTest.txt, contains the results of the test program testmulcmatrixbycvector.c.

• The file, testmulcmatrixbycvector.sh, contains the shell script used to compile, link, and execute the test program testmulcmatrixbycvector.c.

• The file, testmulcvectorbycmatrix.c, contains a test program of Multiply_CVector_by_CMatrix( ) in the file multiply_cvector_by_cmatrix.c. This test program requires the file multiply_cvector_by_cmatrix.c listed above.

• The file, MulCVectorByCMatrixTest.txt, contains the results of the test program testmulcvectorbycmatrix.c.

• The file, testmulcvectorbycmatrix.sh, contains the shell script used to compile, link, and execute the test program testmulcvectorbycmatrix.c.

• The file, testcbilinearpairing.c, contains a test program of CBilinear_Pairing( ) in the file cbilinear_pairing.c. This test program requires the file cbilinear_pairing.c listed above.

• The file, CBilinearPairingTest.txt, contains the results of the test program testcbilinearpairing.c.

• The file, testcbilinearpairing.sh, contains the shell script used to compile, link, and execute the test program testcbilinearpairing.c.

• The file, testcbilinearform.c, contains a test program of CBilinear_Form( ) in the file cbilinear_form.c. This test program requires the file cbilinear_form.c listed above.

• The file, CBilinearFormTest.txt, contains the results of the test program testcbilinearform.c.

• The file, testcbilinearform.sh, contains the shell script used to compile, link, and execute the test program testcbilinearform.c.

• The file, testsesquilinearpairing.c, contains a test program of Sesquilinear_Pairing( ) in the file sesquilinear_pairing.c. This test program requires the file sesquilinear_pairing.c listed above.

• The file, SesquilinearPairingTest.txt, contains the results of the test program testsesquilinearpairing.c.

• The file, testsesquilinearpairing.sh, contains the shell script used to compile, link, and execute the test program testsesquilinearpairing.c.

• The file, testsesquilinearform.c, contains a test program of Sesquilinear_Form( ) in the file sesquilinear_form.c. This test program requires the file sesquilinear_form.c listed above.

• The file, SesquilinearFormTest.txt, contains the results of the test program testsesquilinearform.c.

• The file, testsesquilinearform.sh, contains the shell script used to compile, link, and execute the test program testsesquilinearform.c.