# Jordan Matrix

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## Jordan matrix

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## Jordan Matrix

(also Jordan form of a matrix, or simple classical matrix). With every square matrix A = ǀǀ*a _{ik}*ǀǀ

^{n}

_{1}there is associated a class of similar matrices. This class of matrices always includes a matrix of special normal (canonical) Jordan form, named after M. E. C. Jordan. The Jordan form of a certain eighth-order matrix is given in Figure 1.

Figure 1

Special square blocks, enclosed by broken lines in Figure 1, lie on the principal diagonal. All elements of the matrix that lie outside these blocks are equal to zero. The (complex) numbers forming the principal diagonal of a block are the same; for example, in the first block they are all equal to λ_{1}, and in the second to λ_{2}. The elements of the diagonal immediately above the principal diagonal equal to unity. All other elements in the diagonal blocks are equal to zero. The matrix in Figure 1 contains three diagonal blocks, of which the first is of order 4, and the second and third of order 2. In the general case, the number of blocks and their orders are arbitrary. It is also possible for some of the numbers λ_{1}, λ_{2}, … to be equal. The initial matrix *A* in this example has the elementary divisors (λ – λ_{1})^{4}, (λ – λ_{2})^{2}, and (λ – λ_{3})^{2}. A Jordan matrix is uniquely determined by its elementary divisors.

If the matrix *A* has the Jordan form *I*, then there exists a nonsingular matrix *T* such that *A* = *TIT*^{–1}. Replacement of matrix *A* by the similar Jordan matrix *I* is called a reduction of matrix *A* to normal Jordan form.

To get an idea of the uses of the Jordan form of a matrix, we consider a system of linear differential equations with constant coefficients:

*dx*_{1}/*dt* = *a*_{11}*x*_{1} + *a*_{12}*x*_{2} + … + *a*_{1n}*x _{n}*

*dx*_{2}/*dt* = *a*_{21}*x*_{1} + *a*_{22}*x*_{2} + … + *a*_{2n}*x _{n}*

*dx _{n}*/

*dt*=

*a*

_{n1}

*x*

_{1}+

*a*

_{n2}

*x*

_{2}+ … +

*a*

_{nn}

*x*

_{n}In matrix notation we have

(1) *dx*/*dt* = *Ax*

Let us introduce new unknown functions *y*_{1}, *y*_{2}, …, *y _{n}* by using a nonsingular matrix T = ǀǀ

*t*ǀǀ

_{ik}^{n}

_{1}where the

*t*are numbers (

_{ik}*i, k*= 1, 2, …,

*n*):

*x*_{1} = *t*_{11}*y*_{1} + *t*_{12}*y*_{2} + … + *t*_{1n}*y _{n}*

*x*_{1} = *t*_{21}*y*_{1} + *t*_{22}*y*_{2} + … + *t*_{2n}*y _{n}*

*x _{n}* =

*t*

_{n1}

*y*

_{1}+

*t*

_{n2}

*y*

_{2}+ … +

*t*

_{nn}

*y*

_{n}or in matrix notation

*x* = *Ty*

Substituting this expression for *x* in (1), we obtain

(2) *dy*/*dt* = *Iy*

where matrix *I* is related to matrix *A* by the equation

*A* = *TIT*^{–1}

The matrix *T* is usually selected in such a way that matrix *A* is a Jordan matrix. In this case, the system of equations (2) is much simpler than system (1). For example, if matrix *A* = ǀǀ*a _{ik}*ǀǀ

^{8}

_{1}(

*n*= 8) has the Jordan form given in Figure 1, then system (2) will have the form

*dy*_{1}/*dt* = λ_{1}*y*_{1} + *y*_{2}*dy*_{5}/*dt* = λ_{2}*y*_{5} + *y*_{6}

*dy*_{2}/*dt* = λ_{1}*y*_{2} + *y*_{3}*dy*_{6}/*dt =* λ_{2}*y*_{6}

*dy*_{3}/*dt* = λ_{1}*y*_{3} + *y*_{4}*dy*_{7}/*dt* = λ_{3}*y*_{7} + *y*_{8}

*dy*_{4}/*dt =* λ_{1}*y*_{4}*dy*_{8}/*dt =* λ_{3}*y*_{8}

Integration of this system reduces to a number of integrations of single differential equations.