Jordan Matrix

(also Jordan form of a matrix, or simple classical matrix). With every square matrix A = ǀǀa_ikǀǀⁿ₁ 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 λ₁, and in the second to λ₂. 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 λ₁, λ₂, … to be equal. The initial matrix A in this example has the elementary divisors (λ – λ₁)⁴, (λ – λ₂)², and (λ – λ₃)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:

dx₁/dt = a₁₁x₁ + a₁₂x₂ + … + a_1nx_n

dx₂/dt = a₂₁x₁ + a₂₂x₂ + … + a_2nx_n

dx_n/dt = a_n1x₁ + a_n2x₂ + … + a_nnx_n

In matrix notation we have

(1) dx/dt = Ax

Let us introduce new unknown functions y₁, y₂, …, y_n by using a nonsingular matrix T = ǀǀt_ikǀǀⁿ₁ where the t_ik are numbers (i, k = 1, 2, …, n):

x₁ = t₁₁y₁ + t₁₂y₂ + … + t_1ny_n

x₁ = t₂₁y₁ + t₂₂y₂ + … + t_2ny_n

x_n = t_n1y₁ + t_n2y₂ + … + t_nny_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 = ǀǀa_ikǀǀ⁸₁ (n = 8) has the Jordan form given in Figure 1, then system (2) will have the form

dy₁/dt = λ₁y₁ + y₂dy₅/dt = λ₂y₅ + y₆

dy₂/dt = λ₁y₂ + y₃dy₆/dt = λ₂y₆

dy₃/dt = λ₁y₃ + y₄dy₇/dt = λ₃y₇ + y₈

dy₄/dt = λ₁y₄dy₈/dt = λ₃y₈

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