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 matrices. ...

, a defective matrix is a

square matrix In mathematics, a square matrix is a 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. Square matrices are often ...

that does not have a complete

basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...

eigenvector 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 denoted b ...

s, and is therefore not

diagonalizable In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...

. In particular, an ''n'' × ''n''

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

is defective

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

it does not have ''n''

linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...

eigenvectors. A complete basis is formed by augmenting the eigenvectors with

generalized eigenvector In linear algebra, a generalized eigenvector of an n\times n matrix A is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector. Let V be an n-dimensional vector space; let \phi be a linear map ...

s, which are necessary for solving defective systems of

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...

s and other problems. An ''n'' × ''n'' defective matrix always has fewer than ''n'' distinct

eigenvalue 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 denoted b ...

s, since distinct eigenvalues always have linearly independent eigenvectors. In particular, a defective matrix has one or more eigenvalues ''λ'' with

algebraic multiplicity 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 denoted ...

''m'' > 1 (that is, they are multiple

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 focusing ...

of the characteristic polynomial), but fewer than ''m'' linearly independent eigenvectors associated with ''λ''. If the algebraic multiplicity of ''λ'' exceeds its

geometric multiplicity 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 denoted ...

(that is, the number of linearly independent eigenvectors associated with ''λ''), then ''λ'' is said to be a defective eigenvalue. However, every eigenvalue with algebraic multiplicity ''m'' always has ''m'' linearly independent generalized eigenvectors. A

Hermitian matrix In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -t ...

(or the special case of a

real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...

symmetric matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with ...

) or a

unitary matrix In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if U^* U = UU^* = UU^ = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose is ...

is never defective; more generally, a

normal matrix In mathematics, a complex square matrix is normal if it commutes with its conjugate transpose : The concept of normal matrices can be extended to normal operators on infinite dimensional normed spaces and to normal elements in C*-algebras. As ...

(which includes Hermitian and unitary as special cases) is never defective.

Jordan block

Any nontrivial

Jordan block In the mathematical discipline of matrix theory, a Jordan matrix, named after Camille Jordan, is a block diagonal matrix over a ring (whose identities are the zero 0 and one 1), where each block along the diagonal, called a Jordan block, has t ...

of size 2 × 2 or larger (that is, not completely diagonal) is defective. (A diagonal matrix is a special case of the Jordan normal form with all trivial Jordan blocks and is not defective.) For example, the ''n'' × ''n'' Jordan block :

J = \begin
\lambda & 1            & \;     & \;  \\
\;        & \lambda    & \ddots & \;  \\
\;        & \;           & \ddots & 1   \\
\;        & \;           & \;     & \lambda       
\end,

has an

, λ, with algebraic multiplicity ''n'' (or greater if there are other Jordan blocks with the same eigenvalue), but only one distinct eigenvector

J v_1 = \lambda v_1

, where

v_1 = \begin 1 \\ 0 \\ \vdots \\ 0 \end.

The other canonical basis vectors

v_2 = \begin 0 \\ 1 \\ \vdots \\ 0 \end, ~ \ldots, ~ v_n = \begin 0 \\ 0 \\ \vdots \\ 1 \end

form a chain of generalized eigenvectors such that

J v_k = \lambda v_k + v_

for

k=2,\ldots,n

. Any defective matrix has a nontrivial

Jordan normal form In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF), is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to so ...

, which is as close as one can come to diagonalization of such a matrix.

Example

A simple example of a defective matrix is :

\begin 3 & 1 \\ 0 & 3 \end,

which has a double

of 3 but only one distinct eigenvector :

\begin 1 \\ 0 \end

(and constant multiples thereof).

Notes

References

* * {{Matrix classes Linear algebra

Jordan block

Example

See also

Notes

References