Matrix Polynomials, Minimum Polynomial

Minimum Polynomial

If x is an n×n matrix, unravel x and picture it as a vector in n² space. Similarly, x² looks like a vector in n² space, and so do all the higher powers of x. The first n²+1 powers of x define n²+1 vectors in n² space, and are linearly dependent. A linear combination of these vectors yields 0. In other words, x is the root of a polynomial with degree n²+1 or less.

Of course the coefficients of this polynomial could be fractions of R. We can fix this by multiplying through by a common denominator, but instead, let's just extend R to include its fractions. Now we can divide through by the lead coefficient, whence the polynomial becomes monic.

The minimum polynomial of x is monic, and has minimal degree. If there are two such polynomials, subtract them to find a polynomial of lesser degree. Thus the minimum polynomial is well defined for any x. (If x is 0 let its polynomial be 0.)

Let m(x) be the minimum polynomial, and let p(x) = 0. Use synthetic division to divide m(x) into p(x). If there is a remainder r(x), then r(x) = 0, and r(x) has degree less than that of m(x), which is impossible. Therefore m(x) is a factor of p(x).

Base Change

Let q be a nonsingular matrix that implements a base change. In other words, y = qx/q. We already showed yⁿ is the base change of xⁿ. Verify that base change commutes with addition. Put this all together and p(y) is the base change of p(x), just as p(x) is the inverse base change of p(y). One is 0 iff the other is 0. Therefore similar matrices are roots of the same polynomials, and exhibit the same minimum polynomial.

Eigen Values

convert x to a triangular matrix, with its eigen values down the main diagonal. The diagonal of p(x) is now p applied to the diagonal elements. These must be 0 if p(x) = 0. In other words, the eigen values are roots of p, whenever p(x) = 0.

Block Diagonal

Take another step and convert x to jordan canonical form. Now p(x) is p applied to each block. In particular, p(x) is 0 iff p is 0 on each block. Since R is a field, the polynomials over R form a ufd, and the minimum polynomial m(x) is the least common multiple of the minimum polynomials for each block.