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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the spectrum of a
matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...
is the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of its
eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
s. More generally, if T\colon V \to V is a
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
on any
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
, its spectrum is the set of scalars \lambda such that T-\lambda I is not
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
. The
determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
of the matrix equals the product of its eigenvalues. Similarly, the trace of the matrix equals the sum of its eigenvalues. From this point of view, we can define the pseudo-determinant for a
singular matrix A singular matrix is a square matrix that is not invertible, unlike non-singular matrix which is invertible. Equivalently, an n-by-n matrix A is singular if and only if determinant, det(A)=0. In classical linear algebra, a matrix is called ''non- ...
to be the product of its nonzero eigenvalues (the density of
multivariate normal distribution In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One d ...
will need this quantity). In many applications, such as
PageRank PageRank (PR) is an algorithm used by Google Search to rank web pages in their search engine results. It is named after both the term "web page" and co-founder Larry Page. PageRank is a way of measuring the importance of website pages. Accordin ...
, one is interested in the dominant eigenvalue, i.e. that which is largest in
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
. In other applications, the smallest eigenvalue is important, but in general, the whole spectrum provides valuable information about a matrix.


Definition

Let ''V'' be a finite-dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
over some field ''K'' and suppose ''T'' : ''V'' → ''V'' is a linear map. The ''spectrum'' of ''T'', denoted σ''T'', is the
multiset In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the ''multiplicity'' of ...
of
roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusin ...
of the
characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...
of ''T''. Thus the elements of the spectrum are precisely the eigenvalues of ''T'', and the multiplicity of an eigenvalue ''λ'' in the spectrum equals the dimension of the generalized eigenspace of ''T'' for ''λ'' (also called the algebraic multiplicity of ''λ''). Now, fix a basis ''B'' of ''V'' over ''K'' and suppose ''M'' ∈ Mat''K''(''V'') is a matrix. Define the linear map ''T'' : ''V'' → ''V'' pointwise by ''Tx'' = ''Mx'', where on the right-hand side ''x'' is interpreted as a column vector and ''M'' acts on ''x'' by
matrix multiplication In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...
. We now say that ''x'' ∈ ''V'' is an
eigenvector In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by ...
of ''M'' if ''x'' is an eigenvector of ''T''. Similarly, λ ∈ ''K'' is an eigenvalue of ''M'' if it is an eigenvalue of ''T'', and with the same multiplicity, and the spectrum of ''M'', written σ''M'', is the multiset of all such eigenvalues.


Related notions

The eigendecomposition (or spectral decomposition) of a
diagonalizable matrix In linear algebra, a square matrix A is called diagonalizable or non-defective if it is matrix similarity, similar to a diagonal matrix. That is, if there exists an invertible matrix P and a diagonal matrix D such that . This is equivalent to ...
is a
decomposition Decomposition is the process by which dead organic substances are broken down into simpler organic or inorganic matter such as carbon dioxide, water, simple sugars and mineral salts. The process is a part of the nutrient cycle and is ess ...
of a diagonalizable matrix into a specific canonical form whereby the matrix is represented in terms of its eigenvalues and eigenvectors. The
spectral radius ''Spectral'' is a 2016 Hungarian-American military science fiction action film co-written and directed by Nic Mathieu. Written with Ian Fried (screenwriter), Ian Fried & George Nolfi, the film stars James Badge Dale as DARPA research scientist Ma ...
of a
square matrix In mathematics, a square matrix is a Matrix (mathematics), matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Squ ...
is the largest absolute value of its eigenvalues. In
spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operator (mathematics), operators in a variety of mathematical ...
, the spectral radius of a
bounded linear operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector ...
is the
supremum In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
of the absolute values of the elements in the spectrum of that operator.


Notes


References

* * * * Matrix theory {{Linear-algebra-stub