| diags2_c |
|
Table of contents
Procedure
diags2_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_Input
symmat 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_Output
diag,
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'.
diags2_c chooses `rotate' so that its angle of rotation has
the smallest possible magnitude. If there are two rotations
that meet these criteria (they will be inverses of one
another), either rotation may be chosen.
ParametersNone. Exceptions
Error 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. Particulars
The 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.
Examples
1) 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;
diags2_c ( symmat, diag, rotate );
will return diag, an array containing the eigenvalues of
symmat, and rotate, the coordinate transformation required
to diagonalize symmat. In this case,
diag[0][0] = 28.
diag[1][0] = 0.
diag[0][1] = 0.
diag[1][1] = 2.
and
rotate[0][0] = 0.980580675690920
rotate[1][0] = 0.196116135138184
rotate[0][1] = -0.196116135138184
rotate[1][1] = 0.980580675690920
The columns of rotate give the ellipse's axes, after scaling
them by
1 1
---------------- and ---------------
____________ ____________
\/ diag[0][0] \/ diag[1][1]
respectively.
If smajor and sminor are semi-major and semi-minor axes,
we can find them as shown below. For brevity, we omit the
check for zero or negative eigenvalues.
for ( i = 0; i < 2; i++ )
{
smajor[i] = rotate[i][0] / sqrt( diag[0][0] );
sminor[i] = rotate[i][1] / sqrt( diag[1][1] );
}
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