We define a discrete matrix function called the discrete matrix delayed exponential for two delays , and for two commuting constant matrices B, C as follows.
Definition 2 Let B, C be constant matrices with the property and let , be fixed integers. We define a discrete matrix function called the discrete matrix delayed exponential for two delays m, n and for two constant matrices B, C:
where
(5)
The main property of is given by the following theorem.
Theorem 2 Let B, C be constant matrices with the property and let , be fixed integers. Then
(6)
holds for .
Proof Let . From (1) and (5), we can see easily that, for an integer satisfying
the relation
holds in accordance with Definition 2 of . Since , we have
(7)
Considering the increment by its definition, i.e.,
(8)
we conclude that it is reasonable to divide the proof into four parts with respect to the value of integer k. In case one, k is such that
in case two
in case three
and in case four
We see that the above cases cover all the possible relations between k, and .
In the proof, we use the identities
(9)
where and
(10)
where , which are derived from (2) and (9).
I.
From (1) and (5), we get
Therefore, and, by Definition 2,
(11)
Similarly, omitting details, we get (using (1), and (5)) and
(12)
Let . We show that
(13)
In accordance with (1),
or
From the last inequality, we get
and (13) holds by (2). For that reason and since , we can replace by in (11). Thus, we have
(14)
It is easy to see that, due to (3), formula (14) can be used instead of (11) if also.
Let . Similarly, we can show that
and, since , we can replace by in (12). Thus, we have
(15)
It is easy to see that, due to (3), formula (15) can be used instead of (12) if , too.
Due to (1), we also conclude that
(16)
because
and
The second formula can be proved similarly.
Now we are able to prove that
(17)
With the aid of (7), (8), (9) and (16), we get
By (10), we have
Now in the first sum we replace the summation index i by and in the second sum we replace the summation index j by . Then
Due to (14) and (15), we conclude that formula (17) is valid.
II.
In this case,
and . In addition to this (see relevant computations performed in case I), we have and .
Then
and
(18)
(19)
Like with the computations performed in the previous part of the proof, we get
and
So, we can substitute by in (18) and by in (19).
Accordingly, we have
(20)
(21)
It is easy to see that, due to (3), formula (20) can also be used instead of (18) if and formula (21) can also be used instead of (19) if .
We have to prove
(22)
Therefore,
With the aid of the equation , we get
and, by (9), we have
By (10), we have
Now we replace in the first sum the summation index i by and in the second sum we replace the summation index j by . Then
For , we have
where
Thus,
and formula (22) is proved.
III.
In this case, we have (see relevant computations in cases I and II)
and
Then
and
(23)
(24)
Like with the computations performed in case I, we can get
and
So, we can substitute for in (23) and for in (24).
Thus, we have
(25)
(26)
It is easy to see that, due to (3), formula (25) can also be used instead of (23) if and formula (26) can also be used instead of (24) if .
Now we have to prove
(27)
Considering the difference by its definition, we get
With the aid of relation , we get
and
By (10), we have
Now we replace in the first sum the summation index i by and in the second sum we replace the summation index j by . Then
For , we have
where
Thus,
and formula (27) is proved.
IV.
In this case, we have (see similar combinations in cases II and III)
and
Then
and
(28)
(29)
As before,
and
So, we can substitute for in (28) and for in (29) and
(30)
(31)
It is easy to see that, due to (3), formula (30) can also be used instead of (28) if and formula (31) can also be used instead of (29) if .
Now it is possible to prove the formula
(32)
By definition, we get
With the aid of equations , , we get
and
By (10), we have
We replace in the first sum the summation index i by and in the second sum we substitute the summation index j by . Then
Because , we can express and in the form
where
Thus,
Therefore, formula (32) is valid.
We proved that formula (6) holds in each of the considered cases I, II, III and IV for . If , the proof can be done directly because , ,
and
Formula (6) holds again. Theorem 2 is proved. □