Let A be an mxn matrix whose entries are polynomials with rational coefficients, and let d be the largest integer that occurs as the degree of an entry of A. The representation of A in `long format' is an (mn + 2)x(d + 1) matrix of integers whose rows are: The first row is m followed by d zeros. The second row is the least common denominator of all the coefficients of all the polynomials in A, call it k, followed by d zeros. The remaining mn rows contain the numerators of the d + 1 coefficients of the entries of A (when they are written as fractions with denominator k), reading first from top to bottom in the first column of A, then from top to bottom in the second column of A, and so forth. (Note: The entries are listed one column after another, not one row after another as one might expect.)
For example, the first matrix in the examples section (page 86) is represented in long format by the 4x4 matrix whose first row is 1, 0, 0, 0 (to indicate that the matrix of polynomials has one row), whose second row is 1, 0, 0, 0 (to indicate that 1 is the least common denominator of the coefficients, that is, the coefficients are all integers) and whose last two rows are 0, 1, 2, 1 and 1, 1, 1, 1 (giving the polynomial in the first column and then the polynomial in the second column).
For another example, the third matrix from the last on page 88 (the matrix that has middle row 0, -2, -2) is represented by the 11x4 matrix whose first row is 3, 0, 0, 0, whose second row is 2, 0, 0, 0, and whose remaining 9 rows give the numerators of the coefficients of the 9 entries of the matrix of polynomials, reading column by column, when the denominator is 2. Thus, the third row is 0, 0, 0, 2, the next three rows are entirely zero, the next row is 0, 0, 0, -4, the row after that is 1, -9, 22, -16, the next row is zero, the next row is 0, 0, 0, -4, and the last is 0, 2, -14, 20.
You will be relieved to learn that you will rarely, if ever, have to deal directly with matrices in long format, because functions are provided that will enable you to enter, display, and manipulate matrices whose entries are polynomials without ever needing to know the details about the matrices of integers that represent them in long format.
Return to Chapter 8.