In this section, we will present the maximal and minimal ranks of the linear matrix expression
(4)
where , are given matrices, with respect to four variant matrices . Applying the formula (1) in Lemma 1.1 to the linear matrix expression in (4) and simplifying, we obtain the following result.
Theorem 2.1 Let P(V1, V2, V3, V4) be given as (4). Then
(5)
where
Proof. It is easy to verify by block Gaussian elimination that the rank of P(V1,V2,V3, V4) in (4) can be expressed as
where and , are denotes the identity matrix of order p
i
and q
i
, respectively.
and
According to this result, we have
(6)
Then applying the formula (1) in Lemma 1.1 to matrix T, we have
where
Again applying the formula (1) in Lemma 1.1, we get
(7)
and
(8)
(9)
(10)
(11)
Substituting (8)-(11) into (7) and (6) yield (5).
Recall a simple fact that a matrix equation AXB = C is consistent for every variant matrices X, if and only if the maximal rank of C - AXB with respect to X is zero. Thus, by Theorem 2.1 we can immediately obtain the following result.
Corollary 2.2 Let P(V1,V2,V3, V4) be given as (4). Then the matrix equation A = B1V1C1 + B2V2C2 + B3V3C3 + B4V4C4 holds for any V1, V2, V3, and V4 if and only if T1 = O or T2 = O or T3 = O or T4 = O.
Because the right side of (5) are just composed by ranks of block matrices, they can be easily simplified by block Gaussian elimination when the given matrices in (4) satisfy some restrictions.
Theorem 2.3 Let P(V1, V2, V3, V4) be given as (4) and let R(B1) ⊆ R(B2), R(B3) ⊆ R(B4), Then
(12)
where
Proof. In fact, we can write B1 = B2X, B3 = B4Y, C2 = ZC1, and C4 = WC3 under the hypotheses of Theorem 2.3. In this case, we have
(13)
and
(14)
and
(15)
and
(16)
Combining (5) with (13)-(16) yields (12).
Corollary 2.4 Let P(V1, V2, V3, V4) be given as (4) and let then the matrix equation A = B1V1C1 + B2V2C2 + B3V3C
3
+ B4V4C4 holds for any V1, V2, V3, and V4 if and only if τ1 = O or τ2 = O or τ3 = O.
In the rest of this section, we will find the minimal rank of the linear matrix expression P(V1,V2, V3, V4) in (4), with respect to four variant matrices , when P(V1, V2, V3, V4) satisfy some restrictions.
Theorem 2.5 Let P(V1, V2, V3, V4) be given as (4) and let . Then
(17)
where
Proof. From the proof of Theorem 2.1, it is easy to verify that the minimal rank of P(V1, V2, V3, V4) in (4) can be expressed as
(18)
where T, S, E
i
, p
i
and q
i
, i = 1,2,3,4, are given as the proof of Theorem 2.1. Then applying the formula (2) in Lemma 1.1 to matrix T, we have
(19)
In this case, we derive from (19) that
(20)
Again applying the formula (2) in Lemma 1.1, we have
(21)
where S3 is given as the Equation (7) of the proof of Theorem 2.1. Since B1 = B2X, B3 = B4Y, C2 = ZC1, and C4 = WC3, (21) is reduced to
(22)
The last equality holds, since the well-known Frobenius rank inequality r(ABC) ≥ r(AB) + r(BC) − r(B), then
With the similar method, we also have
(23)
(24)
(25)
On the other hand, by the formula (1) in Lemma 1.1, we have
(26)
(27)