diags2_c |

## Procedurevoid diags2_c ( ConstSpiceDouble symmat [2][2], SpiceDouble diag [2][2], SpiceDouble rotate [2][2] ) ## AbstractDiagonalize a symmetric 2x2 matrix. ## Required_ReadingROTATION ## KeywordsELLIPSE MATRIX ROTATION TRANSFORMATION ## Brief_I/OVariable I/O Description -------- --- -------------------------------------------------- symmat I A symmetric 2x2 matrix. diag O A diagonal matrix similar to symmat. rotate O A rotation used as the similarity transformation. ## Detailed_Inputsymmat A symmetric 2x2 matrix. That is, symmat has the form +- -+ | A B | | |. | B C | +- -+ This routine uses only the upper-triangular elements of symmat, that is, the elements symmat[0][0] symmat[0][1] symmat[1][1] to determine the outputs diag and rotate. ## Detailed_Outputdiag, rotate are, respectively, a diagonal matrix and a 2x2 rotation matrix that satisfy the equation T diag = rotate * symmat * rotate. In other words, diag is similar to symmat, and rotate is a change-of-basis matrix that diagonalizes symmat. ## ParametersNone. ## ExceptionsError free. 1) The matrix element symmat[1][0] is not used in this routine's computations, so the condition symmat[0][1] != symmat[1][0] has no effect on this routine's outputs. ## FilesNone. ## ParticularsThe capability of diagonalizing a 2x2 symmetric matrix is especially useful in a number of geometric applications involving quadratic curves such as ellipses. Such curves are described by expressions of the form 2 2 A x + B xy + C y + D x + E y + F = 0. Diagonalization of the matrix +- -+ | A B/2 | | | | B/2 C | +- -+ allows us to perform a coordinate transformation (a rotation, specifically) such that the equation of the curve becomes 2 2 P u + Q v + R u + S v + T = 0 in the transformed coordinates. This form is much easier to handle. If the quadratic curve in question is an ellipse, we can easily find its center, semi-major axis, and semi-minor axis from the second equation. Ellipses turn up frequently in navigation geometry problems; for example, the limb and terminator (if we treat the Sun as a point source) of a body modelled as a tri-axial ellipsoid are ellipses. A mathematical note: because symmat is symmetric, we can ALWAYS find an orthogonal similarity transformation that diagonalizes symmat, and we can choose the similarity transformation to be a rotation matrix. By `orthogonal' we mean that if the rotate is the matrix in question, then T T rotate rotate = rotate rotate = I. The reasons this routine handles only the 2x2 case are: first, the 2x2 case is much simpler than the general case, in which iterative diagonalization methods must be used, and second, the 2x2 case is adequate for solving problems involving ellipses in 3 dimensional space. Finally, this routine can be used to support a routine that solves the general-dimension diagonalization problem for symmetric matrices. Another feature of the routine that might provoke curiosity is its insistence on choosing the diagonalization matrix that rotates the original basis vectors by the smallest amount. The rotation angle of rotate is of no concern for most applications, but can be important if this routine is used as part of an iterative diagonalization method for higher-dimensional matrices. In that case, it is most undesirable to interchange diagonal matrix elements willy-nilly; the matrix to be diagonalized could get ever closer to being diagonal without converging. Choosing the smallest rotation angle precludes this possibility. ## Examples1) A case that can be verified by hand computation: Suppose symmat is +- -+ | 1.0 4.0 | | | | 4.0 -5.0 | +- -+ Then symmat is similar to the diagonal matrix +- -+ | 3.0 0.0 | | | | 0.0 -7.0 | +- -+ so diag[0][0] = 3. diag[1][0] = 0. diag[0][1] = 0. diag[1][1] = -7. and rotate is +- -+ | 0.89442719099991588 -0.44721359549995794 | | | | 0.44721359549995794 0.89442719099991588 | +- -+ which is an approximation to +- -+ | .4 * 5**(1/2) -.2 * 5**(1/2) | | | | .2 * 5**(1/2) .4 * 5**(1/2) | +- -+ 2) Suppose we want to find the semi-axes of the ellipse defined by 2 2 27 x + 10 xy + 3 y = 1 We can write the above equation as the matrix equation +- -+ +- -+ +- -+ | x y | | 27 5 | | x | = 1 +- -+ | | | | | 5 3 | | y | +- -+ +- -+ Let symmat be the symmetric matrix on the left. The code fragment symmat[0][0] = 27.0; symmat[1][0] = 5.0; symmat[0][1] = 5.0; symmat[1][1] = 3.0; ## RestrictionsNone. ## Literature_References[1] Calculus, Vol. II. Tom Apostol. John Wiley & Sons, 1969. See Chapter 5, `Eigenvalues of Operators Acting on Euclidean Spaces'. ## Author_and_InstitutionN.J. Bachman (JPL) ## Version-CSPICE Version 1.0.0, 13-JUL-1999 (NJB) ## Index_Entriesdiagonalize symmetric 2x2_matrix |

Wed Apr 5 17:54:31 2017