Eigenvalues and eigenvectors
An eigenvector of a square matrix is a non-zero
vector that, when the matrix is multiplied
by, yields a constant multiple of, the multiplier
being commonly denoted by.
That is:
(Because this equation uses post-multiplication
by, it describes a right eigenvector.)
The number is called the eigenvalue of corresponding
to.
If 2D space is visualized as a piece of cloth
being stretched by the matrix, the eigenvectors
would make up the line along the direction
the cloth is stretched in and the line of
cloth at the center of the stretching, whose
direction isn't changed by the stretching
either.
The eigenvalues for the first line would give
the scale to which the cloth is stretched,
and for the second line the scale to which
it's tightened.
A reflection may be viewed as stretching a
line to scale -1 while shrinking the axis
of reflection to scale 1.
For 3D rotations, the eigenvectors form the
axis of rotation, and since the scale of the
axis is unchanged by the rotation, their eigenvalues
are all 1.
In analytic geometry, for example, a three-element
vector may be seen as an arrow in three-dimensional
space starting at the origin.
In that case, an eigenvector is an arrow whose
direction is either preserved or exactly reversed
after multiplication by.
The corresponding eigenvalue determines how
the length of the arrow is changed by the
operation, and whether its direction is reversed
or not, determined by whether the eigenvalue
is negative or positive.
In abstract linear algebra, these concepts
are naturally extended to more general situations,
where the set of real scalar factors is replaced
by any field of scalars (such as algebraic
or complex numbers); the set of Cartesian
vectors is replaced by any vector space (such
as the continuous functions, the polynomials
or the trigonometric series), and matrix multiplication
is replaced by any linear operator that maps
vectors to vectors (such as the derivative
from calculus).
In such cases, the "vector" in "eigenvector"
may be replaced by a more specific term, such
as "eigenfunction", "eigenmode", "eigenface",
or "eigenstate".
Thus, for example, the exponential function
is an eigenfunction of the derivative operator
" ", with eigenvalue, since its derivative
is.
The set of all eigenvectors of a matrix (or
linear operator), each paired with its corresponding
eigenvalue, is called the eigensystem of that
matrix.
Any multiple of an eigenvector is also an
eigenvector, with the same eigenvalue.
An eigenspace of a matrix is the set of all
eigenvectors with the same eigenvalue, together
with the zero vector.
An eigenbasis for is any basis for the set
of all vectors that consists of linearly independent
eigenvectors of.
Not every matrix has an eigenbasis, but every
symmetric matrix does.
The terms characteristic vector, characteristic
value, and characteristic space are also used
for these concepts.
The prefix eigen- is adopted from the German
word eigen for "self-" or "unique to", "peculiar
to", or "belonging to" in the sense of "idiosyncratic"
in relation to the originating matrix.
Eigenvalues and eigenvectors have many applications
in both pure and applied mathematics.
They are used in matrix factorization, in
quantum mechanics, and in many other areas.
Definition
Eigenvectors and eigenvalues of a real matrix
In many contexts, a vector can be assumed
to be a list of real numbers (called elements),
written vertically with brackets around the
entire list, such as the vectors u and v below.
Two vectors are said to be scalar multiples
of each other (also called parallel or collinear)
if they have the same number of elements,
and if every element of one vector is obtained
by multiplying each corresponding element
in the other vector by the same number (known
as a scaling factor, or a scalar).
For example, the vectors
are scalar multiples of each other, because
each element of is −20 times the corresponding
element of.
A vector with three elements, like or above,
may represent a point in three-dimensional
space, relative to some Cartesian coordinate
system.
It helps to think of such a vector as the
tip of an arrow whose tail is at the origin
of the coordinate system.
In this case, the condition " is parallel
to " means that the two arrows lie on the
same straight line, and may differ only in
length and direction along that line.
If we multiply any square matrix with rows
and columns by such a vector, the result will
be another vector, also with rows and one
column.
That is,
where, for each index,
In general, if are not all zeros, the vectors
and will not be parallel.
When they are parallel (that is, when there
is some real number such that ) we say that
is an eigenvector of.
In that case, the scale factor is said to
be the eigenvalue corresponding to that eigenvector.
In particular, multiplication by a 3×3 matrix
may change both the direction and the magnitude
of an arrow in three-dimensional space.
However, if is an eigenvector of with eigenvalue,
the operation may only change its length,
and either keep its direction or flip it (make
the arrow point in the exact opposite direction).
Specifically, the length of the arrow will
increase if, remain the same if, and decrease
it if.
Moreover, the direction will be precisely
the same if, and flipped if.
If, then the length of the arrow becomes zero.
An example
For the transformation matrix
the vector
is an eigenvector with eigenvalue 2.
Indeed,
On the other hand the vector
is not an eigenvector, since
and this vector is not a multiple of the original
vector.
Another example
For the matrix
we have
Therefore, the vectors and are eigenvectors
of corresponding to the eigenvalues 1 and
3 respectively.
(Here the symbol indicates matrix transposition,
in this case turning the row vectors into
column vectors.)
Trivial cases
The identity matrix (whose general element
is 1 if, and 0 otherwise) maps every vector
to itself.
Therefore, every vector is an eigenvector
of, with eigenvalue 1.
More generally, if is a diagonal matrix (with
whenever ), and is a vector parallel to axis
(that is,, and if ), then where.
That is, the eigenvalues of a diagonal matrix
are the elements of its main diagonal.
This is trivially the case of any 1 ×1 matrix.
General definition
The concept of eigenvectors and eigenvalues
extends naturally to abstract linear transformations
on abstract vector spaces.
Namely, let be any vector space over some
field of scalars, and let be a linear transformation
mapping into.
We say that a non-zero vector of is an eigenvector
of if (and only if) there is a scalar in such
that
This equation is called the eigenvalue equation
for, and the scalar is the eigenvalue of corresponding
to the eigenvector.
Note that means the result of applying the
operator to the vector, while means the product
of the scalar by.
The matrix-specific definition is a special
case of this abstract definition.
Namely, the vector space is the set of all
column vectors of a certain size ×1, and
is the linear transformation that consists
in multiplying a vector by the given matrix.
Some authors allow to be the zero vector in
the definition of eigenvector.
This is reasonable as long as we define eigenvalues
and eigenvectors carefully: If we would like
the zero vector to be an eigenvector, then
we must first define an eigenvalue of as a
scalar in such that there is a nonzero vector
in with.
We then define an eigenvector to be a vector
in such that there is an eigenvalue in with.
This way, we ensure that it is not the case
that every scalar is an eigenvalue corresponding
to the zero vector.
Eigenspace and spectrum
If is an eigenvector of, with eigenvalue,
then any scalar multiple of with nonzero is
also an eigenvector with eigenvalue, since.
Moreover, if and are eigenvectors with the
same eigenvalue and, then is also an eigenvector
with the same eigenvalue.
Therefore, the set of all eigenvectors with
the same eigenvalue, together with the zero
vector, is a linear subspace of, called the
eigenspace of associated to.
If that subspace has dimension 1, it is sometimes
called an eigenline.
The geometric multiplicity of an eigenvalue
is the dimension of the eigenspace associated
to, i.e. number of linearly independent eigenvectors
with that eigenvalue.
The eigenspaces of T always form a direct
sum (and as a consequence any family of eigenvectors
for different eigenvalues is always linearly
independent).
Therefore the sum of the dimensions of the
eigenspaces cannot exceed the dimension n
of the space on which T operates, and in particular
there cannot be more than n distinct eigenvalues.
Any subspace spanned by eigenvectors of is
an invariant subspace of, and the restriction
of T to such a subspace is diagonalizable.
The set of eigenvalues of is sometimes called
the spectrum of.
Eigenbasis
An eigenbasis for a linear operator that operates
on a vector space is a basis for that consists
entirely of eigenvectors of (possibly with
different eigenvalues).
Such a basis exists precisely if the direct
sum of the eigenspaces equals the whole space,
in which case one can take the union of bases
chosen in each of the eigenspaces as eigenbasis.
The matrix of T in a given basis is diagonal
precisely when that basis is an eigenbasis
for T, and for this reason T is called diagonalizable
if it admits an eigenbasis.
Generalizations to infinite-dimensional spaces
The definition of eigenvalue of a linear transformation
remains valid even if the underlying space
is an infinite dimensional Hilbert or Banach
space.
Namely, a scalar is an eigenvalue if and only
if there is some nonzero vector such that.
Eigenfunctions
A widely used class of linear operators acting
on infinite dimensional spaces are the differential
operators on function spaces.
Let be a linear differential operator in on
the space of infinitely differentiable real
functions of a real argument.
The eigenvalue equation for is the differential
equation
The functions that satisfy this equation are
commonly called eigenfunctions of.
For the derivative operator, an eigenfunction
is a function that, when differentiated, yields
a constant times the original function.
The solution is an exponential function
including when is zero when it becomes a constant
function.
Eigenfunctions are an essential tool in the
solution of differential equations and many
other applied and theoretical fields.
For instance, the exponential functions are
eigenfunctions of the shift operators.
This is the basis of Fourier transform methods
for solving problems.
Spectral theory
If is an eigenvalue of, then the operator
is not one-to-one, and therefore its inverse
does not exist.
The converse is true for finite-dimensional
vector spaces, but not for infinite-dimensional
ones.
In general, the operator may not have an inverse,
even if is not an eigenvalue.
For this reason, in functional analysis one
defines the spectrum of a linear operator
as the set of all scalars for which the operator
has no bounded inverse.
Thus the spectrum of an operator always contains
all its eigenvalues, but is not limited to
them.
Associative algebras and representation theory
More algebraically, rather than generalizing
the vector space to an infinite dimensional
space, one can generalize the algebraic object
that is acting on the space, replacing a single
operator acting on a vector space with an
algebra representation – an associative
algebra acting on a module.
The study of such actions is the field of
representation theory.
A closer analog of eigenvalues is given by
the representation-theoretical concept of
weight, with the analogs of eigenvectors and
eigenspaces being weight vectors and weight
spaces.
Eigenvalues and eigenvectors of matrices
Characteristic polynomial
The eigenvalue equation for a matrix is
which is equivalent to
where is the identity matrix.
It is a fundamental result of linear algebra
that an equation has a non-zero solution if,
and only if, the determinant of the matrix
is zero.
It follows that the eigenvalues of are precisely
the real numbers that satisfy the equation
The left-hand side of this equation can be
seen (using Leibniz' rule for the determinant)
to be a polynomial function of the variable.
The degree of this polynomial is, the order
of the matrix.
Its coefficients depend on the entries of,
except that its term of degree is always.
This polynomial is called the characteristic
polynomial of; and the above equation is called
the characteristic equation (or, less often,
the secular equation) of.
For example, let be the matrix
The characteristic polynomial of is
which is
The roots of this polynomial are 2, 1, and
11.
Indeed these are the only three eigenvalues
of, corresponding to the eigenvectors and
(or any non-zero multiples thereof).
In the real domain
Since the eigenvalues are roots of the characteristic
polynomial, an matrix has at most eigenvalues.
If the matrix has real entries, the coefficients
of the characteristic polynomial are all real;
but it may have fewer than real roots, or
no real roots at all.
For example, consider the cyclic permutation
matrix
This matrix shifts the coordinates of the
vector up by one position, and moves the first
coordinate to the bottom.
Its characteristic polynomial is which has
one real root.
Any vector with three equal non-zero elements
is an eigenvector for this eigenvalue.
For example,
In the complex domain
The fundamental theorem of algebra implies
that the characteristic polynomial of an matrix,
being a polynomial of degree, has exactly
complex roots.
More precisely, it can be factored into the
product of linear terms,
where each is a complex number.
The numbers,,..., (which may not be all distinct)
are roots of the polynomial, and are precisely
the eigenvalues of.
Even if the entries of are all real numbers,
the eigenvalues may still have non-zero imaginary
parts (and the elements of the corresponding
eigenvectors will therefore also have non-zero
imaginary parts).
Also, the eigenvalues may be irrational numbers
even if all the entries of are rational numbers,
or all are integers.
However, if the entries of are algebraic numbers
(which include the rationals), the eigenvalues
will be (complex) algebraic numbers too.
The non-real roots of a real polynomial with
real coefficients can be grouped into pairs
of complex conjugate values, namely with the
two members of each pair having the same real
part and imaginary parts that differ only
in sign.
If the degree is odd, then by the intermediate
value theorem at least one of the roots will
be real.
Therefore, any real matrix with odd order
will have at least one real eigenvalue; whereas
a real matrix with even order may have no
real eigenvalues.
In the example of the 3×3 cyclic permutation
matrix, above, the characteristic polynomial
has two additional non-real roots, namely
where is the imaginary unit.
Note that,, and.
Then
Therefore, the vectors and are eigenvectors
of, with eigenvalues, and, respectively.
Algebraic multiplicities
Let be an eigenvalue of an matrix.
The algebraic multiplicity of is its multiplicity
as a root of the characteristic polynomial,
that is, the largest integer such that divides
evenly that polynomial.
Like the geometric multiplicity, the algebraic
multiplicity is an integer between 1 and;
and the sum of over all distinct eigenvalues
also cannot exceed.
If complex eigenvalues are considered, is
exactly.
It can be proved that the geometric multiplicity
of an eigenvalue never exceeds its algebraic
multiplicity.
Therefore, is at most.
If, then is said to be a semisimple eigenvalue.
Example
For the matrix:
The roots of this polynomial, and hence the
eigenvalues, are 2 and 3.
The algebraic multiplicity of each eigenvalue
is 2; in other words they are both double
roots.
On the other hand, the geometric multiplicity
of the eigenvalue 2 is only 1, because its
eigenspace is spanned by the vector, and is
therefore 1-dimensional.
Similarly, the geometric multiplicity of the
eigenvalue 3 is 1 because its eigenspace is
spanned by.
Hence, the total algebraic multiplicity of
A, denoted, is 4, which is the most it could
be for a 4 by 4 matrix.
The geometric multiplicity is 2, which is
the smallest it could be for a matrix which
has two distinct eigenvalues.
Diagonalization and eigendecomposition
If the sum of the geometric multiplicities
of all eigenvalues is exactly, then has a
set of linearly independent eigenvectors.
Let be a square matrix whose columns are those
eigenvectors, in any order.
Then we will have, where is the diagonal matrix
such that is the eigenvalue associated to
column of.
Since the columns of are linearly independent,
the matrix is invertible.
Premultiplying both sides by we get.
By definition, therefore, the matrix is diagonalizable.
Conversely, if is diagonalizable, let be a
non-singular square matrix such that is some
diagonal matrix.
Multiplying both sides on the left by we get.
Therefore each column of must be an eigenvector
of, whose eigenvalue is the corresponding
element on the diagonal of.
Since the columns of must be linearly independent,
it follows that.
Thus is equal to if and only if is diagonalizable.
If is diagonalizable, the space of all -element
vectors can be decomposed into the direct
sum of the eigenspaces of.
This decomposition is called the eigendecomposition
of, and it is preserved under change of coordinates.
A matrix that is not diagonalizable is said
to be defective.
For defective matrices, the notion of eigenvector
can be generalized to generalized eigenvectors,
and that of diagonal matrix to a Jordan form
matrix.
Over an algebraically closed field, any matrix
has a Jordan form and therefore admits a basis
of generalized eigenvectors, and a decomposition
into generalized eigenspaces
Further properties
Let be an arbitrary matrix of complex numbers
with eigenvalues,,....
(Here it is understood that an eigenvalue
with algebraic multiplicity occurs times in
this list.)
Then
The trace of, defined as the sum of its diagonal
elements, is also the sum of all eigenvalues:
The determinant of is the product of all eigenvalues:
The eigenvalues of the th power of, i.e. the
eigenvalues of, for any positive integer,
are
The matrix is invertible if and only if all
the eigenvalues are nonzero.
If is invertible, then the eigenvalues of
are
If is equal to its conjugate transpose (in
other words, if is Hermitian), then every
eigenvalue is real.
The same is true of any a symmetric real matrix.
If is also positive-definite, positive-semidefinite,
negative-definite, or negative-semidefinite
every eigenvalue is positive, non-negative,
negative, or non-positive respectively.
Every eigenvalue of a unitary matrix has absolute
value.
Left and right eigenvectors
The use of matrices with a single column (rather
than a single row) to represent vectors is
traditional in many disciplines.
For that reason, the word "eigenvector" almost
always means a right eigenvector, namely a
column vector that must be placed to the right
of the matrix in the defining equation
There may be also single-row vectors that
are unchanged when they occur on the left
side of a product with a square matrix; that
is, which satisfy the equation
Any such row vector is called a left eigenvector
of.
The left eigenvectors of are transposes of
the right eigenvectors of the transposed matrix,
since their defining equation is equivalent
to
It follows that, if is Hermitian, its left
and right eigenvectors are complex conjugates.
In particular if is a real symmetric matrix,
they are the same except for transposition.
Variational characterization
In the Hermitian case, eigenvalues can be
given a variational characterization.
The largest eigenvalue of is the maximum value
of the quadratic form.
A value of that realizes that maximum, is
an eigenvector.
For more information, see Min-max theorem.
Calculation
Computing the eigenvalues
The eigenvalues of a matrix can be determined
by finding the roots of the characteristic
polynomial.
Explicit algebraic formulas for the roots
of a polynomial exist only if the degree is
4 or less.
According to the Abel–Ruffini theorem there
is no general, explicit and exact algebraic
formula for the roots of a polynomial with
degree 5 or more.
It turns out that any polynomial with degree
is the characteristic polynomial of some companion
matrix of order.
Therefore, for matrices of order 5 or more,
the eigenvalues and eigenvectors cannot be
obtained by an explicit algebraic formula,
and must therefore be computed by approximate
numerical methods.
In theory, the coefficients of the characteristic
polynomial can be computed exactly, since
they are sums of products of matrix elements;
and there are algorithms that can find all
the roots of a polynomial of arbitrary degree
to any required accuracy.
However, this approach is not viable in practice
because the coefficients would be contaminated
by unavoidable round-off errors, and the roots
of a polynomial can be an extremely sensitive
function of the coefficients (as exemplified
by Wilkinson's polynomial).
Efficient, accurate methods to compute eigenvalues
and eigenvectors of arbitrary matrices were
not known until the advent of the QR algorithm
in 1961.
Combining the Householder transformation with
the LU decomposition results in an algorithm
with better convergence than the QR algorithm.
For large Hermitian sparse matrices, the Lanczos
algorithm is one example of an efficient iterative
method to compute eigenvalues and eigenvectors,
among several other possibilities.
Computing the eigenvectors
Once the (exact) value of an eigenvalue is
known, the corresponding eigenvectors can
be found by finding non-zero solutions of
the eigenvalue equation, that becomes a system
of linear equations with known coefficients.
For example, once it is known that 6 is an
eigenvalue of the matrix
we can find its eigenvectors by solving the
equation, that is
This matrix equation is equivalent to two
linear equations
Both equations reduce to the single linear
equation.
Therefore, any vector of the form, for any
non-zero real number, is an eigenvector of
with eigenvalue.
The matrix above has another eigenvalue.
A similar calculation shows that the corresponding
eigenvectors are the non-zero solutions of,
that is, any vector of the form, for any non-zero
real number.
Some numeric methods that compute the eigenvalues
of a matrix also determine a set of corresponding
eigenvectors as a by-product of the computation.
History
Eigenvalues are often introduced in the context
of linear algebra or matrix theory.
Historically, however, they arose in the study
of quadratic forms and differential equations.
In the 18th century Euler studied the rotational
motion of a rigid body and discovered the
importance of the principal axes.
Lagrange realized that the principal axes
are the eigenvectors of the inertia matrix.
In the early 19th century, Cauchy saw how
their work could be used to classify the quadric
surfaces, and generalized it to arbitrary
dimensions.
Cauchy also coined the term racine caractéristique
(characteristic root) for what is now called
eigenvalue; his term survives in characteristic
equation.
Fourier used the work of Laplace and Lagrange
to solve the heat equation by separation of
variables in his famous 1822 book Théorie
analytique de la chaleur.
Sturm developed Fourier's ideas further and
brought them to the attention of Cauchy, who
combined them with his own ideas and arrived
at the fact that real symmetric matrices have
real eigenvalues.
This was extended by Hermite in 1855 to what
are now called Hermitian matrices.
Around the same time, Brioschi proved that
the eigenvalues of orthogonal matrices lie
on the unit circle, and Clebsch found the
corresponding result for skew-symmetric matrices.
Finally, Weierstrass clarified an important
aspect in the stability theory started by
Laplace by realizing that defective matrices
can cause instability.
In the meantime, Liouville studied eigenvalue
problems similar to those of Sturm; the discipline
that grew out of their work is now called
Sturm–Liouville theory.
Schwarz studied the first eigenvalue of Laplace's
equation on general domains towards the end
of the 19th century, while Poincaré studied
Poisson's equation a few years later.
At the start of the 20th century, Hilbert
studied the eigenvalues of integral operators
by viewing the operators as infinite matrices.
He was the first to use the German word eigen
which means "own", to denote eigenvalues and
eigenvectors in 1904, though he may have been
following a related usage by Helmholtz.
For some time, the standard term in English
was "proper value", but the more distinctive
term "eigenvalue" is standard today.
The first numerical algorithm for computing
eigenvalues and eigenvectors appeared in 1929,
when Von Mises published the power method.
One of the most popular methods today, the
QR algorithm, was proposed independently by
John G.F.
Francis and Vera Kublanovskaya in 1961.
Applications
Eigenvalues of geometric transformations
The following table presents some example
transformations in the plane along with their
2×2 matrices, eigenvalues, and eigenvectors.
Note that the characteristic equation for
a rotation is a quadratic equation with discriminant,
which is a negative number whenever is not
an integer multiple of 180°.
Therefore, except for these special cases,
the two eigenvalues are complex numbers,;
and all eigenvectors have non-real entries.
Indeed, except for those special cases, a
rotation changes the direction of every nonzero
vector in the plane.
Schrödinger equation
An example of an eigenvalue equation where
the transformation is represented in terms
of a differential operator is the time-independent
Schrödinger equation in quantum mechanics:
where, the Hamiltonian, is a second-order
differential operator and, the wavefunction,
is one of its eigenfunctions corresponding
to the eigenvalue, interpreted as its energy.
However, in the case where one is interested
only in the bound state solutions of the Schrödinger
equation, one looks for within the space of
square integrable functions.
Since this space is a Hilbert space with a
well-defined scalar product, one can introduce
a basis set in which and can be represented
as a one-dimensional array and a matrix respectively.
This allows one to represent the Schrödinger
equation in a matrix form.
The bra–ket notation is often used in this
context.
A vector, which represents a state of the
system, in the Hilbert space of square integrable
functions is represented by.
In this notation, the Schrödinger equation
is:
where is an eigenstate of.
It is a self adjoint operator, the infinite
dimensional analog of Hermitian matrices (see
Observable).
As in the matrix case, in the equation above
is understood to be the vector obtained by
application of the transformation to.
Molecular orbitals
In quantum mechanics, and in particular in
atomic and molecular physics, within the Hartree–Fock
theory, the atomic and molecular orbitals
can be defined by the eigenvectors of the
Fock operator.
The corresponding eigenvalues are interpreted
as ionization potentials via Koopmans' theorem.
In this case, the term eigenvector is used
in a somewhat more general meaning, since
the Fock operator is explicitly dependent
on the orbitals and their eigenvalues.
If one wants to underline this aspect one
speaks of nonlinear eigenvalue problem.
Such equations are usually solved by an iteration
procedure, called in this case self-consistent
field method.
In quantum chemistry, one often represents
the Hartree–Fock equation in a non-orthogonal
basis set.
This particular representation is a generalized
eigenvalue problem called Roothaan equations.
Geology and glaciology
In geology, especially in the study of glacial
till, eigenvectors and eigenvalues are used
as a method by which a mass of information
of a clast fabric's constituents' orientation
and dip can be summarized in a 3-D space by
six numbers.
In the field, a geologist may collect such
data for hundreds or thousands of clasts in
a soil sample, which can only be compared
graphically such as in a Tri-Plot (Sneed and
Folk) diagram, or as a Stereonet on a Wulff
Net.
The output for the orientation tensor is in
the three orthogonal (perpendicular) axes
of space.
The three eigenvectors are ordered by their
eigenvalues; then is the primary orientation/dip
of clast, is the secondary and is the tertiary,
in terms of strength.
The clast orientation is defined as the direction
of the eigenvector, on a compass rose of 360°.
Dip is measured as the eigenvalue, the modulus
of the tensor: this is valued from 0° (no
dip) to 90° (vertical).
The relative values of,, and are dictated
by the nature of the sediment's fabric.
If, the fabric is said to be isotropic.
If, the fabric is said to be planar.
If, the fabric is said to be linear.
Principal components analysis
The eigendecomposition of a symmetric positive
semidefinite (PSD) matrix yields an orthogonal
basis of eigenvectors, each of which has a
nonnegative eigenvalue.
The orthogonal decomposition of a PSD matrix
is used in multivariate analysis, where the
sample covariance matrices are PSD.
This orthogonal decomposition is called principal
components analysis (PCA) in statistics.
PCA studies linear relations among variables.
PCA is performed on the covariance matrix
or the correlation matrix (in which each variable
is scaled to have its sample variance equal
to one).
For the covariance or correlation matrix,
the eigenvectors correspond to principal components
and the eigenvalues to the variance explained
by the principal components.
Principal component analysis of the correlation
matrix provides an orthonormal eigen-basis
for the space of the observed data: In this
basis, the largest eigenvalues correspond
to the principal-components that are associated
with most of the covariability among a number
of observed data.
Principal component analysis is used to study
large data sets, such as those encountered
in data mining, chemical research, psychology,
and in marketing.
PCA is popular especially in psychology, in
the field of psychometrics.
In Q methodology, the eigenvalues of the correlation
matrix determine the Q-methodologist's judgment
of practical significance (which differs from
the statistical significance of hypothesis
testing; cf. criteria for determining the
number of factors).
More generally, principal component analysis
can be used as a method of factor analysis
in structural equation modeling.
Vibration analysis
Eigenvalue problems occur naturally in the
vibration analysis of mechanical structures
with many degrees of freedom.
The eigenvalues are used to determine the
natural frequencies (or eigenfrequencies)
of vibration, and the eigenvectors determine
the shapes of these vibrational modes.
In particular, undamped vibration is governed
by
that is, acceleration is proportional to position
(i.e., we expect to be sinusoidal in time).
In dimensions, becomes a mass matrix and a
stiffness matrix.
Admissible solutions are then a linear combination
of solutions to the generalized eigenvalue
problem
where is the eigenvalue and is the angular
frequency.
Note that the principal vibration modes are
different from the principal compliance modes,
which are the eigenvectors of alone.
Furthermore, damped vibration, governed by
leads to what is called a so-called quadratic
eigenvalue problem,
This can be reduced to a generalized eigenvalue
problem by clever use of algebra at the cost
of solving a larger system.
The orthogonality properties of the eigenvectors
allows decoupling of the differential equations
so that the system can be represented as linear
summation of the eigenvectors.
The eigenvalue problem of complex structures
is often solved using finite element analysis,
but neatly generalize the solution to scalar-valued
vibration problems.
Eigenfaces
In image processing, processed images of faces
can be seen as vectors whose components are
the brightnesses of each pixel.
The dimension of this vector space is the
number of pixels.
The eigenvectors of the covariance matrix
associated with a large set of normalized
pictures of faces are called eigenfaces; this
is an example of principal components analysis.
They are very useful for expressing any face
image as a linear combination of some of them.
In the facial recognition branch of biometrics,
eigenfaces provide a means of applying data
compression to faces for identification purposes.
Research related to eigen vision systems determining
hand gestures has also been made.
Similar to this concept, eigenvoices represent
the general direction of variability in human
pronunciations of a particular utterance,
such as a word in a language.
Based on a linear combination of such eigenvoices,
a new voice pronunciation of the word can
be constructed.
These concepts have been found useful in automatic
speech recognition systems, for speaker adaptation.
Tensor of moment of inertia
In mechanics, the eigenvectors of the moment
of inertia tensor define the principal axes
of a rigid body.
The tensor of moment of inertia is a key quantity
required to determine the rotation of a rigid
body around its center of mass.
Stress tensor
In solid mechanics, the stress tensor is symmetric
and so can be decomposed into a diagonal tensor
with the eigenvalues on the diagonal and eigenvectors
as a basis.
Because it is diagonal, in this orientation,
the stress tensor has no shear components;
the components it does have are the principal
components.
Eigenvalues of a graph
In spectral graph theory, an eigenvalue of
a graph is defined as an eigenvalue of the
graph's adjacency matrix, or (increasingly)
of the graph's Laplacian matrix (see also
Discrete Laplace operator), which is either
(sometimes called the combinatorial Laplacian)
or (sometimes called the normalized Laplacian),
where is a diagonal matrix with equal to the
degree of vertex, and in, the th diagonal
entry is.
The th principal eigenvector of a graph is
defined as either the eigenvector corresponding
to the th largest or th smallest eigenvalue
of the Laplacian.
The first principal eigenvector of the graph
is also referred to merely as the principal
eigenvector.
The principal eigenvector is used to measure
the centrality of its vertices.
An example is Google's PageRank algorithm.
The principal eigenvector of a modified adjacency
matrix of the World Wide Web graph gives the
page ranks as its components.
This vector corresponds to the stationary
distribution of the Markov chain represented
by the row-normalized adjacency matrix; however,
the adjacency matrix must first be modified
to ensure a stationary distribution exists.
The second smallest eigenvector can be used
to partition the graph into clusters, via
spectral clustering.
Other methods are also available for clustering.
Basic reproduction number
The basic reproduction number () is a fundamental
number in the study of how infectious diseases
spread.
If one infectious person is put into a population
of completely susceptible people, then is
the average number of people that one typical
infectious person will infect.
The generation time of an infection is the
time,, from one person becoming infected to
the next person becoming infected.
In a heterogeneous population, the next generation
matrix defines how many people in the population
will become infected after time has passed.
is then the largest eigenvalue of the next
generation matrix.
