We must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants by fixing upon a number p, and selecting at will p lines and p columns, the squares corresponding to which may be termed determinants of the path order.

- J. J. SYLVESTER, Additions to the Articles, "On a New Class of Theorems," and "On Pascal's Theorem" (1850)

I have in previous papers defined a "Matrix" as a rectangular array of terms, out of which different systems of determinants may be engendered, as from the womb of a common parent; these cognate determinants being by no means isolated in their relations to one another, but subject to certain simple laws of mutual dependence and simultaneous deperition.

- J. J. SYLVESTER, On the Relation Between the Minor Determinants of Linearly Equivalent Quadratic Functions (1851)

Overview

The linear matrix equation

where , and are given and is to be determined, is called the Sylvester equation. It is of pedagogical interest because it includes as special cases several important linear equation problems:

1. linear system: Ax = c,

2. multiple right-hand side linear system: AX = C,

3. matrix inversion: AX = I,

4. eigenvector corresponding to given eigenvalue b: ( A ? bI) x = 0,