matrix_det (complex)

Computes the determinant of a square, complex matrix.

Synopsis

#include <imsl.h>

f_complex imsl_c_matrix_det (int n, f_complex a[], ..., 0)

The type d_complex function is imsl_z_matrix_det.

Required Arguments

int n (Input)
The number of rows and columns in matrix A.

float a[] (Input)
Matrix for which the determinant is calculated.

Return Value

The determinant of a. If IMSL_RETURN_LOG is specified, the log of the absolute value of the determinant. If the return value cannot be computed, NaN is returned.

Synopsis with Optional Arguments

#include <imsl.h>

f_complex imsl_c_matrix_det (int n, f_complex a[],

IMSL_HERMITIAN,

IMSL_TOLERANCE, float tol,

IMSL_RETURN_LOG, f_complex *s,

0)

Optional Arguments

IMSL_HERMITIAN,
If specified, the input matrix is expected to be Hermitian, and this property will be exploited in the calculation of the eigenvalues.

IMSL_TOLERANCE, float tol (Input)
Scalar containing the tolerance used to determine when an eigenvalue is negligible and replaced by the value zero. A zero or negligible eigenvalue implies that the input matrix is singular and has zero determinant.

Default: tol = 100.0 * imsl_f_machine(4)

IMSL_RETURN_LOG, f_complex *s (Output)
If specified, the log of the modulus of the determinant log(|det A|)) is returned in the return value and the sign is returned in the variable s. The determinant is recovered by det(A) = s*exp log(|det A|)), where the multiplication is for complex numbers. See example 3, below.

Description

Let A be a complex-valued square matrix with n rows and columns. The determinant of A, denoted by det(A) or |A|, is a scalar function that characterizes the matrix and its behavior as a linear operator. There are several ways to calculate the determinant. Based on the characteristic equation that is used to find the eigenvalues of a matrix, it follows that

where λi are the n (real or complex) eigenvalues of A. This function first finds the eigenvalues and then takes their product to obtain the determinant of A. See imsl_c_eig_gen and imsl_c_eig_herm for further details.

Example 1

The following example finds the determinant of a 4x4 complex matrix.

#include <imsl.h>
#include <stdio.h>

int main()
{   
   f_complex deta = {1.0, 0.0};
   int n = 4;

   f_complex a[] = {
     {5.0, 9.0}, {5.0, 5.0}, {-6.0, -6.0}, {-7.0, -7.0},
     {3.0, 3.0}, {6.0, 10.0}, {-5.0, -5.0}, {-6.0, -6.0},
     {2.0, 2.0}, {3.0, 3.0}, {-1.0, 3.0}, {-5.0, -5.0},
     {1.0, 1.0}, {2.0, 2.0}, {-3.0, -3.0}, {0.0, 4.0} };
   
   deta = imsl_c_matrix_det(n, a, 0);
 
   imsl_c_write_matrix("Example 1 determinant:", 1, 1, &deta, 0);

}

Output

Example 1 determinant:
(        400,      -2160)

Example 2

The following example finds the determinant of a 3x3 complex Hermitian matrix.

#include <imsl.h>
#include <stdio.h>

int main()
{
       int n = 3;
       f_complex deta = { 1.0,0.0 };
    f_complex a[] = {
            {1.0, 0.0}, {1.0, -7.0}, {0.0, -1.0},
            {1.0, 7.0}, {5.0, 0.0}, {10.0, -3.0},
            {0.0, 1.0}, {10.0, 3.0}, {-2.0, 0.0}};


     
    deta = imsl_c_matrix_det(n, a, IMSL_HERMITIAN, 0);
   
    imsl_c_write_matrix("Example 2 determinant:", 1, 1, &deta, 0);  
    
}

Output

Example 2 determinant:
(        122,          0)

Example 3

The following example finds the determinant of a 3x3 and a 1x1 complex matrix using the option, IMSL_LOG_RETURN.

#include <imsl.h>
#include <stdio.h>

int main()
{
 f_complex ldeta = { 1.0, 0.0 };
 f_complex deta = { 0.0, 0.0 };
 f_complex s;
 int n = 3;

 f_complex a[9] = {
     {1.0, 2.0}, {3.0, 4.0}, {21.0, 22.0},
     {43.0, 44.0}, { 13.0, 14.0}, {15.0, 16.0},
     {5.0, 6.0}, {7.0, 8.0}, {25.0, 26.0} };

 ldeta = imsl_c_matrix_det(n, a, IMSL_RETURN_LOG, &s, 0);
 deta.re = exp(ldeta.re);
 deta = imsl_c_mul(deta, s);

 
 imsl_c_write_matrix("Example 3a log determinant", 1, 1, &ldeta, 0);
 imsl_c_write_matrix("Example 3a determinant", 1, 1, &deta, 0);

 
 n = 1;
 ldeta = imsl_c_matrix_det(n, a, IMSL_RETURN_LOG, &s, 0);
 deta.re = exp(ldeta.re);
 deta.im = 0.0;
 deta = imsl_c_mul(deta, s);

 imsl_c_write_matrix("Example 3b log determinant", 1, 1, &ldeta, 0);
 imsl_c_write_matrix("Example 3b determinant", 1, 1, &deta, 0);


}

Output

Example 3a log determinant
   (      8.725,      0.000)

Example 3a determinant
 (      -4352,       4352)

Example 3b log determinant
   (     0.8047,     0.0000)

Example 3b determinant
 (          1,          2)

Fatal Errors