diags2_c |

Table of contents## Procedurediags2_c ( Diagonalize symmetric 2x2 matrix ) void 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 is 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 modeled 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] T. Apostol, "Calculus, Vol. II," chapter 5, "Eigenvalues of Operators Acting on Euclidean Spaces," John Wiley & Sons, 1969. ## Author_and_InstitutionN.J. Bachman (JPL) J. Diaz del Rio (ODC Space) ## Version-CSPICE Version 1.0.1, 17-JUN-2021 (JDR) Edited the header to comply with NAIF standard. -CSPICE Version 1.0.0, 13-JUL-1999 (NJB) ## Index_Entriesdiagonalize symmetric 2x2_matrix |

Fri Dec 31 18:41:04 2021