38. Partial matrices

Let there be given the rs-matrices A_ik (i = 1, ··· , r; k = 1, ··· , s) of the type (m_i,n_k). Arranging these matrices in rows and columns, you obtain

of the type (m,n), where

m = m₁ + m₂ + ··· + m_r,
n = n₁ + n₂ + ··· + n_s.

Conversely, you can decompose a given matrix A in many ways into such partial matrices. However, we will only consider decompositions in which partial matrices, located side by side, have equally many rows and those in columsn equally many columns. The types of the individual partial matrices are called partial types. The decomposition of a matrix into its rows (s = 1; all m_i=1) or columns (r = 1; all n_k=1) are special cases which have already been encountered earlier.

Two matrices of the same type, which have been decomposed into partial matrices in the same manner, are obviously added by addition of the corresponding partial matrices.

Now, let there be given a second matrix B of the type (m, p), which has also been decomposed into partial matrices as follows:

where B_kl is of the type (n_k,p_l). The partial types of A and B are then said to be interlinked. As is readily confirmed, you can then form the product AB as if the A_ik and B_kl were not matrices but numbers, whence

with

A special role have those decompositions of square matrices, in which splitting in horizontal and vertical directions occurs in the same manner, that is, r = s, m_i = n_i (i = 1, ··· , r). The diagonal components A_ii are then squares. The matrix A is said to be decomposable if you can decompose it into partial matrices so that above (or below) the diagonal squares are only nul-matrices, that is,

If B has been decomposed in a corresponding manner, the same holds for A+B and AB, and the diagonal squares of A+B and AB are the sums A_ii + B_ii and A_iiB_ii, respectively.

If there are also below the diagonal squares nul-matrices, that is,

the matrix A is said to decompose completely into the digonal components A₁₁,··· , A_rr. Instead of (38.1), we will employ the simpler notation

A = A₁₁ + A₂₂ + ··· + A_rr. (38.2)

If both A and B decompose completely in the same way, so do A+B and AB.

Now follow a few aspects which will become useful below. Since there are in a (m,n)-matrix A of Rank r exactly r linearly independent rows and r linearly independent columns, you can interchange the rows and columns in such a manner that there arises a matrix A₁, in which the first r rows and the first r columns are linearly independent. * The interchange of the rows of A results in multiplication of A from the left by a regular (m,m)-matrix R the rows and columns of which are only a single 1 and otherwise zeroes. This matrix R is called a permutation matrix. If the a_i-th row of A is to move in A₁ into the i-th position, then the 1 must occur in R in the i-th row in the a_i-th position. Correspondingly, the interchange of the columns of A is equivalent to multiplication of A from the right by an (n,n)-permutation-matrix S. Thus, A₁=RAS.

* We have in mind the case r < m, r < n. If r = m or r = n, certain of the changes are not required.

Now, in A₁, the rows with the numbers r + 1, ··· , m and the columns with the numbers r + 1, ··· , n are multiple sums of the first r rows and columns, respectively. Hence you can add such linear combinations of the first r rows (columns) to the latter rows (columns) to create null-rows (-columns). Also these operations arise through multiplication by certain regular matrices from the left and right, respectively. For example, the addition of the j-th row of A₁, multiplied by l, to the i-th row (i j) is equivalent to multiplication from the left of A₁ by an (m,m)-matrix T₁, which arises from the m-row unit matrix by writing l at the intersection of the i-th row and j-th column instead of 0. The addition of the columns corresponds to multiplication from the right by (n,n)-matrices of a corresponding type. At the end of these operations, you arrive from A₁ at an (m,n)-matrix TA₁U of the form

C is here a regular (r,r)-matrix, since it has been assumed that the rows (and columns) of C are linearly independent.

Finally, multiply A₂ from the left by the regular (m,m)-matrix

and obtain

In DA₂ = DTRASU, combine DTR in a regular (m,m)-Matrix Q and SU in a regular (n,n)-matrix P and obtain the

Auxiliary Rule: There always exist regular matrices P and Q with the property

for an (m,n)-matrix of Rank r.

* Of course, in the case r = m, you obtain QAP = (E_mO) and for r = n The case |A| 0 is trivial.

We now introduce a concept of great importance in matrix theory which we will discuss in more detail in the last Section 47. Let there be given two (n,n)-matrices K and L. If there exists a regular matrix P for which K = P^-1LP, K is said to be similar to L. If K is similar to L, then L is also similar to P, whence we can say that K and L are mutually similar, because from K=P^-1LP follows L=(P^-1)^-1KP^-1.

Let M be a set of (n,n) matrices and denote its matrices by A, B, ···.

If P is a regular matrix (which need not be contained in M), we will denote the set of the matrices

P^-1AP,P^-1BP, ···

by P^-1MP and call it a set, similar to M.

A set M is reducible if there exists a set P^-1MP, all matrices of which are decomposable in the same manner, that is,

where all the matrices A₁, B₁, ··· are of the same type (n₁,n₁), and hence also all the A₂, B₂, ··· are of the same type (n₂,n₂). Naturally, you have n₁ + n₂ = n. A set M is irreducible, if there does not exist a set, similar to M, all matrices of which can be decomposed in this manner.

By a matrix V, which can be interchanged with the set M, one understands a matrix which for every matrix M of M satiesfies the relation VM = MV. In a trivial manner, multiples gE of the unit matrix, socalled scalar matrices, are interchangeable.

Schur's auxiliary rule:

Only scalar matrices are interchangeable with an irreducible matrix M of (n,n) matrices.

Proof:

Let V be a matrix which can be interchanged with M. The proof is indirect in that we conclude that M is reducible from the assumption that V is not a scalar matrix.

If V is not a scalar matrix, you can select a number l so that |V - lE| = 0, but W = V - lE O. In fact, if you let l be a characteristic root of V (cf. Section 39), V as well as lE can be interchanged with M, whence the same is true for W = V - lE. Moreover, you have for the Rank r of W the inequality 1 r < n. According to Schur's Rule, there exist then two regular matrices P and Q such that

The equation WM = MW, valid for all matrices M of the set M, yields

or, in more detail,

where

are subdivided into partial matrices so that M and N are of the type (r,r). Completion of the multiplication in (38.3) yields

whence M₂ = 0 and M is reducible, because all matrices of P^-1MP are decomposed in the same manner. Hence Schur's Rule has been proved. we will employ it in the proof of

Theorem 21

Let M be a set of pairwise interchangeable (n,n)-matrices. Then there exists a set, similar to M, which only consists of triangular matrices.

Proof: We will employ induction with respect to the number of rows n. No proof is required for n=1. We assume that Theorem 21 is true for sets of pairwise interchangeable matrices with less than n rows.

If all matrices of M are scalar, the Rule certainly is true. Hence we can assume that there exists in M a matrix which does not have the form lE. According to our assumption, it is interchangeable with all matrices of M. According to Schur's Rule, M is then reducible, that is, there exists a regular matrix P with the property that every matrix P^-1MP of P^-1MP has the form

The matrices M₁, all of which are of the type (r,r) with 1 r < n, form a set M₁ of pairwise interchangeable matrices. The matrices M₄ also form a set M₄ of pairwise interchangeable matrices of the type (n - r, n - r). The fact that (38.4) is valid for every matrix M of M can be expressed by the equation

According to the induction assumption, there exist two regular matrices R₁ and R₂ of the types (r,r) and (n-r,n-r), respectively, with the property that R₁^-1M₁R₁ and R₄^-1M₄R₄ only consist of diagonal matrices. Setting

you find

where M is a set of (n-r,n-r)-matrices, which are of no further interest. Since R₁^-1M₁R₁ as well as R₄^-1M₄R₄ consist only of triangular matrices, the same is true for (PR)^-1M(PR), whence the Theorem has been proved.

If you consider as M a single square matrix, you find:

Every square matrix is similar to a triangular matrix.

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