Global Positioning System Reference
In-Depth Information
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If
A
is positive definite, then (A.57) is the equation of a
u
-dimensional ellipsoid
expressed in a Cartesian coordinate system (x). The center of the ellipsoid is at
x
=
o
.
Transforming (rotating) the coordinate system (x)
x
=
Ey
(A.61)
expresses the quadratic form in the (y) coordinate system,
y
T
E
T
AEy
=
v
(A.62)
Since the matrix
E
consists of normalized eigenvectors we can use (A.47) to obtain
the simple expressions
y
T
v
=
Λ
y
[35
(A.63)
y
1
λ
1
+
y
2
λ
2
+···+
y
u
λ
u
=
Lin
—
0.7
——
Nor
*PgE
This expression can be written as
y
1
v/
y
2
v/
y
u
v/
λ
1
+
λ
2
+ ··· +
λ
u
=
1
(A.64)
This is the equation for the
u
-dimensional ellipsoid in the principal axes form; i.e.,
the coordinate system (y) coincides with the principal axes of the hyperellipsoid, and
the lengths of the principal axes are proportional to the reciprocal of the square root
of the eigenvalues. All eigenvalues are positive because the matrix
A
is positive def-
inite. Equation (A.61) determines the orientation between the (x) and (y) coordinate
systems. If
A
has a rank defect, the dimension of the hyperellipsoid is
R(
A
)
[35
=
r<u
.
Let the vectors
x
and
y
be of dimension
u
×
1 and let the
u
×
u
matrix
A
contain
constants. Consider the quadratic form
x
T
Ay
y
T
A
T
w
=
=
x
(A.65)
Because (A.65) is a 1
1 matrix, the expression can be transposed. This fact is used
frequently to simplify expressions when deriving least-squares solutions. The total
differential
dw
is
×
∂w
∂
x
d
x
∂w
∂
y
d
y
dw
=
+
(A.66)
Th
e vectors
d
x
and
d
y
contain the differentials of the components of
x
and
y
, respec-
tiv
ely. From (A.66) and (A.65) it follows that
y
T
A
T
d
x
x
T
A
d
y
dw
=
+
(A.67)
If the matrix
A
is symmetric, then the total differential of
