The matrix equation AX=XB

Parker, W. V.

doi:10.1215/s0012-7094-50-01705-4

Cited by 10 publications

(4 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Now equations (3.2) and (3.3) have exactly the form studied in the non-derogatory case so the existence and structure of Yn and Note that now not only must the characteristic of F divide n, it must also divide and n2. Equations (3.4) and (3.5), on the other hand, are precisely the form studied by Parker (1950) , it follows that the number of arbitrary parameters in is the deg of f 2. Thus, for the derogatory case, the existence and structure of a solution to equation (3.1) is established.…”

Section: E Ro G a To R Y Casementioning

confidence: 95%

“…The methods used are constructive in nature and thus provide the structure of the matrix X whenever the first part of this question is answered in the affirma tive. These methods were used by Parker (1950) in studying the matrix equation A X = X B . There he used the rational canonical form to great advantage in constructing the matrix X for a given pair, A and B.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Almost commutative matrices

Dixon

Daley

1994

Proc. R. Soc. Lond. A

View full text Add to dashboard Cite

The n × n matrices A and X over a field F are called almost commutative if AX - XA = I . This equation cannot hold if the characteristic of F is either zero or greater than n . In the case where the characteristic of F divides n , certain pairs A and X , exist. It is the purpose of this paper not only to prove the existence of such pairs, but to construct (in terms of arbitrary parameters) the most general matrix X for a given matrix A . The methods use the rational canonical form so as to facilitate constructability.

show abstract

Section: E Ro G a To R Y Casementioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Almost commutative matrices

Dixon

Daley

1994

Proc. R. Soc. Lond. A

View full text Add to dashboard Cite

show abstract

“…showing that any tensor equation of the form (16) is equivalent to a matrix equation of the form (17). Note that the tensor A can be interpreted as a linear map that transforms vectors in…”

Section: Example 32 (Inverse Tensors) Consider the Three Fourth Ordmentioning

confidence: 99%

Stretching and condensing tensors. Application to tensor and matrix equations

Tolosa¹,

Ron²

2006

ijcms

View full text Add to dashboard Cite

Since tensors are vectors, no matter their order and valencies, they can be represented in matrix form and in particular as column matrices of components with respect to their corresponding bases. This consideration allows using all the available tools for vectors and can be extremely advantageous to deal with some interesting problems in linear algebra, as solving tensor linear equations. Once, the tensors have been operated as vectors, they can be returned to their initial notation as tensors. This technique, that is specially useful for implementing computer programs, is illustrated by several examples of applications. In particular, some interesting tensor and matrix equations are solved. Several numerical examples are used to illustrate the proposed methodology.

show abstract

“…As one of the basic linear matrix equation, AX = XB was examined (see, e.g., Hartwig [56], Horn and Johnson [70], Parker [112], Slavova et al [125]). In general cases, solutions of AX = XB can only be determined through canonical forms of A and B.…”

Section: Proof Follows From (41) ✷mentioning

confidence: 99%