In the
mathematical fields of
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrice ...
, a principal axis is a certain line in a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
associated with an
ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a surface that may be defined as th ...
or
hyperboloid
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...
, generalizing the major and minor
axes of an
ellipse
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
or
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
. 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
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrice ...
and
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
, the principal axis theorem is a geometrical counterpart of the
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful be ...
. It has applications to the
statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
of
principal components analysis
Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and ...
and the
singular value decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is re ...
. In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, the theorem is fundamental to the studies of
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
and
birefringence
Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or birefractive). The birefri ...
.
Motivation
The equations in the
Cartesian plane R
2:
:
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-terms'' involving 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
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
or a
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
. The basic observation is that if, by completing the square, the quadratic expression can be reduced to a sum of two squares then the equation defines an ellipse, whereas if it reduces to a difference of two squares then the equation represents a hyperbola:
:
Thus, in our example expression, the problem is how to absorb the coefficient of the cross-term 8''xy'' 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 quadratic form as
:
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
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful be ...
, it has
real eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
and is
diagonalizable by an
orthogonal matrix (''orthogonally diagonalizable'').
To orthogonally diagonalize ''A'', one must first find its eigenvalues, and then find an
orthonormal
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of ...
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'' =
1 u2">''u1 u2is an orthogonal matrix, since it has orthonormal columns, and ''A'' is diagonalized by:
:
This applies to the present problem of "diagonalizing" the quadratic form through the observation that
:
Thus, the equation
is that of an ellipse, since the left side can be written as 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 R
2. 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 ''c''
1 and ''c''
2 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 ''c''
12 + 9''c''
22 = 1 occurs when ''c''
2 = 0, so at the points ''c''
1 = ±1. Similarly, the minimum distance is where ''c''
2 = ±1/3.
It is possible now to read off the major and minor axes of this ellipse. These are precisely the individual
eigenspace
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s of the matrix ''A'', since these are where ''c''
2 = 0 or ''c''
1 = 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
The principal axis theorem concerns
quadratic forms in R
''n'', which are
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
s 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 λ
1, ..., λ
''n'' (possibly repeated according to their
algebraic multiplicities) and the corresponding orthonormal eigenbasis is u
1, ..., u
''n''. Then,
:
and
:
where ''c''
''i'' is the ''i''-th entry of c . Furthermore,
: The ''i''-th principal axis is the line determined by equating ''c''
''j'' =0 for all
. The ''i''-th principal axis is the span of the vector u
''i'' .
See also
*
Sylvester's law of inertia
References
* {{cite book, authorlink=Gilbert Strang, first=Gilbert, last=Strang, title=Introduction to Linear Algebra, publisher=Wellesley-Cambridge Press, year=1994, isbn=0-9614088-5-5
Theorems in geometry
Theorems in linear algebra