As already has been defined in Section 38: An (n-n)-matrix is similar to an (n,n)=matrix B if there exists a regular matrix P such that A = P-1BP. Similarity has the properties:
1. Reflexivity: Every matrix is similar to itself, for
always E-1AE = A.
2. Symmetry: If A is
similar to B, then also B is similar to A,
for it follows from A = P-1BP that B
= P1-1AP1 with
P1 = P-1.
3. Transitivity: If A is similar to B and
B is similar to C, then A is also
similar to C, because A = P-1BP and
B = Q-1BQ yields A = (QP)-1C(QP).
As a consequence of these properties, you obtain as you join mutually similar matrices in a class a subdivision of all (n,n)-Matrices into element free partial sets - classes of similarity.
A quantity which has for all similar matrices the same value is called an invariant of similarity. For example, according to the Rank Equation in Section 24, Rank is an invariant of similarity. Theorem 22 states that also the characteristic roots, the coefficients of the characteristic polynomial and, in particular, the determinant are similarity invariants. We will study the task to find a complete system of invariants of similarity, that is such a system that two matrices are exactly similar if their invariants agree. The fact that Rank and characteristic roots do not yet form a complete system of invariants is demonstrated by the two matrices
Both of them have the same characteristic roots and the same Rank, but are not similar, because due to P-1EP = E, only the unit matrix is similar to itself.
Let A and B be two matrices, both of which decompose in the same manner into diagonal minors
Hence, if corresponding diagonal components are similar, that is,
Ai = Pi-1BiPi (i = 1, 2, ··· , r)
then also A and B are similar, because with
you find A = P-1BP.
When you confine your
considerations to matrices with elements out of a fixed body of
numbers, the following observation is useful: If A and B
are matrices with elements out of K which altogether are similar, there also always exists a
regular matrix P with elements of K for
which A = P-1BP, because this equation can
also be written in the form PA = BP with the auxiliary
condition
|P| 
0. This shows that the determination of P results
in the solution of a system of linear equations. However, every
system of linear equations has already a complete system of
solutions in every body which contains the coefficients.