This is rewritten as
a solution of which is given in Lemma 12. Lemmas 12 (a) and (b) show that, under the assumptions 1 ≠ |λi λj|, (i, j =1, ... , p), for the character- istic roots of B , the solution 11 for the equation (55) is given by (49) and (50).
By the assumption that Σ11 and Σ22 are positive definite, Σ11 + BΣ22B' is positive definite. Therefore, Lemmas 12 (c) and (d) show that the solu- tion 11 is also positive definite, so that ( ) can be regarded as a density function.
To take care about the cases with ≠ 0, ≠ 0, apply the develop- ment above to such cases after the transformation = － , = － . (b) In addition to the development of (a), there is one thing to men- tion. Lemma 11 tells that the characteristic roots v1, ...,vq of B are given by v1 = λ1, ... , vp = λp and vp+1 = 0p+1, ... ,vq = 0q, Therefore, under the assumption 1 >|λi λj|, (i, j =1, ... , p), the condition that 1 > |vivj|, (i, j =1, ... , q), holds. This validates that the development for (a) applies to (b). ■
7.3 Derivation of the joint distribution
Lemma 17 For the parameters of (25) and (26), B and B are assumed to be diagonalizable with the characteristic roots being λ1, ... , λp for B . Furthermore, it is assumed that the characteristic roots satisfy the assumption 1 > |λi λj|, (i, j =1, ... , p), and that Σ11 and Σ22 are positive definite. Then we haυe the propositions (a) and (b) below.
(a) The joint distribution consistent with (25) and (26) is
where 22 is defined by (52) and (53).
Proof. (a) Assume that = 0, = 0. Given (25) and (26). Lemma 16 (a) shows under the assumptions that the marginal distribution of X is ( )
= N(0, 11). Then the product of ( ) and = N( , Σ22) of (26) gives the joint distribution
where
Letting
we have, from Lemma 9 (e),
Based on a proposition concerning matrix determinant, we get
Then can be rewritten
And if we consider a quadratic form of , we have, for a non-zero vector t' = [ r', s' ],
which is positive. So, is a positive definite matrix, and (57) satisfies a condition to be a normal density function. And is (56).
For the case with ≠ 0, ≠ 0, the development above applies after the transformation = － , = － .
(b) Similar to (a), with an additional note mentioned in Proof of Lemma 16 (b). ■
Theorem 18 Assume that the conditional distributions are given as (25) and (26), where B and B are diagonalizable with the character- istic roots being λ1, ... , λp for B . Furthermore, it is assumed that the characteristic roots satisfy the assumption 1 > |λi λj|, (i, j = 1, ... , p), and that Σ11 and Σ22 are positive definite. Still in addition we assume
Then the joint distribution of X and Y consistent with (25) and (26) is = and
Here 11 is a solution for the equation (48) and given as (49) and (50). 22
is a solution for (51) and given as (52) and (53). We also have that 12 = '21 and that 12 is a solution for
Proof. Under the assumption 1 > |λi λj|, (i, j = 1, ... , p), Lemmas 17 (a) and (b) hold and 11 and 22 of the consistent joint distribution satisfy (48) and (51). Multiplying (48) by ' and (51) by B, we get
Recalling, from Lemmas 17 (a) and (b), that 12 = 11 ', 21 = B 22, (59) and (60) are rewritten as
In view of the assumption (58) B Σ22( ≡ Σ12) = Σ11 ', the right hand sides of (61) and (62) respectively can be rewritten as
So that the equations (59) and (60) for 12 and '21 become
The equations (63) and (64) have a common right hand side Σ12 + BΣ12 ', so these are written as one equation
for Z. It is seen that 12 and '21 are the solutions for (65), for which the uniqueness of solution is already shown in Lemmas 12 (a) and (b).
Therefore, we have 12 = '21. ■
7.4 Summary
The results developed so far can be summarized in a way.
The covariance matrix of the consistent joint distribution satisfies (48), (51), (61) and (62). these equations are put into one
This is rewritten in a compact form
where
It is pointed out that the equation (66) for shows a symmetricity in a sense. The characteristic roots of A are those λ1, ... , λp of B as well as those v1, ... , vq of B. We have already shown that (v1, ... , vq) = (λ1, ... , λp, 0p+1, ... , 0q), see Lemma 11. Therefore, under the assumption 1 > |λi λj|, (i, j = 1, ... , p), Lemmas 12 (a), (b), (c) and (d) apply to the equation (66).
We have assumed that B and B are diagonalizable and that their characteristic roots are, respectively, λ1, ... , λp and v1, ... , vq. Lemma 12, based on these assumptions, shows the properties of the matrix equa- tions X－B X B = W and X－B X 'B' = W or X－UXV = W in general.
In theory of Statistics, we do not come across very often a notion of matrix diagonalization except for that of covariance matrices, which are symmetric and diagonalizable. However, the diagonalizability of matri- ces B and B are hard to assume in our context. So, in this section, we remove the assumption of diagonalizable B and B to extend the development of the previous sections.
We assume in this section that the matrices B and B (or U and V in general) can be transformed into Jordan canonical form. And Lemma 12 is extended in this respect.
8.1 Transformation of the matrix equation The equation (28) of interest is repeated here;
where X = (m × n) is an unknown matrix and three known matrices U = (m × m), V = (n × n) and W = (m × n) have their real elements. It is assumed without loss of generality that m ≦ n. Thus, (67) is a matrix equation for X given U, V and W. We will consider the properties of (67) and its solution. The results will be applied to the distribution problem of this paper, as Lemma 12 has been applied so far.
Letting U = (m × m) be a square matrix with real elements, and λ1, ... , λr be its characteristics roots (distinct roots), we assume that U is trans- formed into Jordan canonical form
where T is a non-singular matrix and Jordan canonical form J is expressed as
Here Ji called Jordan cell associated with the characteristic roots λi is expressed as Ji = [ λi ] = (1 × 1) or
with mi being some constants determined by U. Jordan cell Ji, for which mi ≧ 2, can be written as
where
with I = (mi－1 × mi－1). Jordan canonical form (68) composed of Jordan cells J1, J2, ... , Jr is denoted as J = J1 J2 … Jr .
We also assume that V = (n × n) is a square matrix with distinct
4)
4) For Jordan canonical transformation, see, for example, S. L. Lipschutz and M. Lipson (2013) : Linear Algebra, Fifth edition, McGraw Hill, N. Y.
and V respectively with the characteristic roots λ1, ... , λr and v1, ... , vs in (67) are denoted as
where Ji = (mi × mi), (i = 1, ... , r), is Jordan cell associated with λi, and Kj = ( nj × nj), ( j = 1, ... , s), with nj being some constant determined by V, is the one associated with vj.
Now, by the non-singular matrices T and S, the equation (67) is trans- formed into X－TJT－1XSKS－1 = W or
If we let Z = T－1XS and Q = T－1WS, then the equation (67) is rewritten as
where
with Zij = (mi × ni) and Qij = (mi × nj).
Lemma 19 If Z* is a solution for (71), then X* = TZ*S－1, where T and S are any non-singular matrices, is a solution for (67).
Proof. Let Z* be a solution for (71), then Z*－JZ*K= Q. Let X* = TZ*S－1, and substitute X* in the left hand side of (67). Then we have
This shows X* is a solution for (67). ■ The equation (71) can be partitioned
Since Ji and Kj are Jordan cells, (72) is rewritten as
A special case of (73), with mi = 1 and nj = 1, would take a simple form
yielding a solution Zij = (1 × 1) = Qij /(1－λi v j). Note that from now on we only consider the cases where mi > 1 and ni > 1. The method of analysis would be the same for the other cases with mi = 1 or nj = 1.
Removing the subscripts i and j, (73) is written as
If we let the ( , h)－element of Z of (75) be zh, ( = 1, ... , mi, h = 1, ... , nj), and the ( , h)－element of Q be qh, then (75) is rewritten as
where the subscripts i and j for mi, nj, λj and vj are also omitted.
