Jordan Matrix

Jordan Matrix

In the mathematical discipline of matrix theory, a Jordan block over a ring (whose identities are the zero 0 and one 1) is a matrix composed of 0 elements everywhere except for the diagonal, which is filled with a fixed element, and for the superdiagonal, which is composed of ones. The concept is named after Camille Jordan.

begin{pmatrix}
lambda & 1 & 0 & cdots & 0 \
0 & lambda & 1 & cdots & 0 \
vdots & vdots & vdots& ddots & vdots \
0 & 0 & 0 & lambda & 1 \
0 & 0 & 0 & 0 & lambda end{pmatrix}

Every Jordan block is thus specified by its dimension n and its eigenvalue and is indicated as . Any block diagonal matrix whose blocks are Jordan blocks is called a Jordan matrix; using either the or the “” symbol, the block diagonal square matrix whose first diagonal block is, whose second diagonal block is and whose third diagonal block is is compactly indicated as or, respectively. For example the matrix


J=left(begin{matrix}
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & i & 1 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & i & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & i & 1 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & i & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 1 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 1 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 end{matrix}right)

is a Jordan matrix with a block with eigenvalue, two blocks with eigenvalue the imaginary unit and a block with eigenvalue 7. Its Jordan-block structure can also be written as either or .

Read more about Jordan MatrixLinear Algebra, Functions of Matrices, Dynamical Systems, Linear Ordinary Differential Equations

Other articles related to "jordan matrix, matrix":

Jordan Matrix - Linear Ordinary Differential Equations
... let and whose direct closed-form solution involves computation of the matrix exponential Another way, provided the solution is restricted to the local Lebesgue space of -dimensional ... In this case The matrix function is called the resolvent matrix of the differential operator ... It is meromorphic with respect to the complex parameter since its matrix elements are rational functions whose denominator is equal for all to ...

Famous quotes containing the words matrix and/or jordan:

    As all historians know, the past is a great darkness, and filled with echoes. Voices may reach us from it; but what they say to us is imbued with the obscurity of the matrix out of which they come; and try as we may, we cannot always decipher them precisely in the clearer light of our day.
    Margaret Atwood (b. 1939)

    Let me just say, at once: I am not now nor have I ever been a white man. And, leaving aside the joys of unearned privilege, this leaves me feeling pretty good ...
    —June Jordan (b. 1936)