In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
discipline of
matrix theory
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.
For example,
\begi ...
, a Jordan matrix, named after
Camille Jordan
Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''.
Biography
Jordan was born in Lyon and educated at ...
, is a
block diagonal matrix
In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original m ...
over a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
(whose
identities are the
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
0 and
one 1), where each block along the diagonal, called a Jordan block, has the following form:
Definition
Every Jordan block is specified by its dimension ''n'' and its
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 ...
, and is denoted as . It is an
matrix of zeroes everywhere except for the diagonal, which is filled with
and for the
superdiagonal, which is composed of ones.
Any block diagonal matrix whose blocks are Jordan blocks is called a Jordan matrix. This square matrix, consisting of diagonal blocks, can be compactly indicated as
or
, where the ''i''-th Jordan block is .
For example, the matrix
is a Jordan matrix with a block with
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 ...
, two blocks with eigenvalue the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
, and a block with eigenvalue 7. Its Jordan-block structure is written as either
or .
Linear algebra
Any square matrix whose elements are in an
algebraically closed field is
similar to a Jordan matrix , also in
, which is unique up to a permutation of its diagonal blocks themselves. is called the
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 ...
of and corresponds to a generalization of the diagonalization procedure. A
diagonalizable matrix
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.) ...
is similar, in fact, to a special case of Jordan matrix: the matrix whose blocks are all .
More generally, given a Jordan matrix
, that is, whose th diagonal block,
, is the Jordan block and whose diagonal elements
may not all be distinct, the
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 ...
of
for the matrix , indicated as
, corresponds to the number of Jordan blocks whose eigenvalue is . Whereas the index of an eigenvalue
for , indicated as
, is defined as the dimension of the largest Jordan block associated to that eigenvalue.
The same goes for all the matrices similar to , so
can be defined accordingly with respect to the
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 ...
of for any of its eigenvalues
. In this case one can check that the index of
for is equal to its multiplicity as a
root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
of the
minimal polynomial of (whereas, by definition, its
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 ...
for ,
, is its multiplicity as a root of the
characteristic polynomial of ; that is,