Similarity of Matrices
Harold M. Edwards · Birkhäuser Boston eBooks · 1995
In applications of linear algebra, the linear substitutions that arise often describe transformations of an n-dimensional space. A “transformation of an n-dimensional space” is a function that changes a quantity described by n numbers (mathematicians habitually think of such a quantity as a “point” in “n-dimensional space” even when n > 3) into a new quantity of the same kind. A linear substitution with n new variables and n original variables can be thought of as describing such a transformation by giving the relation between the original values of the n numbers and their values after the transformation is applied. The matrix of coefficients of a linear substitution that arises in this way is square. This is one reason that square matrices are especially important.