Principal axis
theorem

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free encyclopedia

In the mathematical fields
of geometry and linear
algebra, a principal
axis is a certain line
in a Euclidean space associated to an ellipsoid or hyperboloid, generalizing
the major and minor axes
of an ellipse. The principal axis theorem states that the principal axes are
perpendicular, and gives a constructive procedure for finding them.

Mathematically, the
principal axis theorem is a generalization of the method of completing
the square from elementary
algebra. In linear
algebra and functional
analysis, the principal axis theorem is a geometrical counterpart of
the spectral theorem. It
has applications to the statistics of principal
components analysis and
the singular
value decomposition. In physics, the
theorem is fundamental to the study of angular momentum.

Contents

• 1 Motivation


• 2 Formal
  statement


• 3 See
  also


• 4 References

Motivation[edit]

The equations in the Cartesian
plane R²:

define, respectively, an
ellipse and a hyperbola. In each case, the x and y axes
are the principal axes. This is easily seen, given that there are no cross-termsinvolving
products xy in
either expression. However, the situation is more complicated for equations
like

Here some method is required
to determine whether this is an ellipse or a hyperbola. The basic observation is
that if, by completing the square, the expression can be reduced to a sum of two
squares then it defines an ellipse, whereas if it reduces to a difference of two
squares then it is the equation of a hyperbola:

Thus, in our example
expression, the problem is how to absorb the coefficient of the cross-term
8xy into the
functions u and v. Formally, this problem is
similar to the problem of matrix
diagonalization, where one tries to find a suitable coordinate system in
which the matrix of a linear transformation is diagonal. The first step is to
find a matrix in which the technique of diagonalization can be applied.

The trick is to write the
equation in the following form:

where the cross-term has
been split into two equal parts. The matrix A in
the above decomposition is a symmetric matrix. In
particular, by the spectral theorem, it
hasreal eigenvalues and is diagonalizable by an orthogonal matrix (orthogonally
diagonalizable).

To orthogonally
diagonalize A, one must
first find its eigenvalues, and then find an orthonormal eigenbasis.
Calculation reveals that the eigenvalues of A are

with corresponding
eigenvectors

Dividing these by their
respective lengths yields an orthonormal eigenbasis:

Now the matrix S = [u₁ u₂]
is an orthogonal matrix, since it has orthonormal columns, and A is diagonalized by:

This applies to the present
problem of "diagonalizing" the equation through the observation that

Thus, the equation is that
of an ellipse, since it is the sum of two squares.

It is tempting to simplify
this expression by pulling out factors of 2. However, it is important not to
do this. The quantities

have a geometrical meaning.
They determine an orthonormal coordinate
system on R². In other words, they are obtained from the
original coordinates by the application of a rotation (and possibly a
reflection). Consequently, one may use the c₁ and c₂ coordinates to make statements
about length and
angles (particularly
length), which would otherwise be more difficult in a different choice of
coordinates (by rescaling them, for instance). For example, the maximum distance
from the origin on the ellipse c₁² + 9c₂² = 1 occurs when c₂=0, so at the points c₁=±1. Similarly, the minimum distance is
where c₂=±1/3.

It is possible now to read
off the major and minor axes of this ellipse. These are precisely the individual
eigenspaces of the matrix A, since these are
where c₂ = 0 orc₁=0.
Symbolically, the principal axes are

To summarize:

• The equation is for an ellipse, since both
  eigenvalues are positive. (Otherwise, if one were positive and the other
  negative, it would be a hyperbola.)


• The principal axes are the lines spanned by
  the eigenvectors.


• The minimum and maximum distances to the
  origin can be read off the equation in diagonal form.

Using this information, it
is possible to attain a clear geometrical picture of the ellipse: to graph it,
for instance.

Formal statement[edit]

The principal
axis theorem concern quadratic
forms in Rⁿ, which are homogeneous
polynomials of degree 2. Any quadratic form may be represented as

where A is a symmetric matrix.

The first part of the
theorem is contained in the following statements guaranteed by the spectral
theorem:

• The eigenvalues of A are real.


• A is
  diagonalizable, and the eigenspaces of A are
  mutually orthogonal.

In particular, A is orthogonally diagonalizable,
since one may take a basis of each eigenspace and apply the Gram-Schmidt
process separately within
the eigenspace to obtain an orthonormal eigenbasis.

For the second part, suppose
that the eigenvalues of A are
λ₁, ..., λ_n (possibly repeated according to their
algebraic multiplicities) and the corresponding orthonormal eigenbasis is u₁,...,u_n. Then

where the c_i are the coordinates with respect to
the given eigenbasis. Furthermore,

• The i-th principal
  axis is the line
  determined by the n-1
  equations c_j = 0, j ≠ i.
  This axis is the span of the vector u_i.

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