Companion Matrices and Their Relations to Toeplitz and Hankel Matrices

In this paper we describe some properties of companion matrices and demonstrate some special patterns that arise when a Toeplitz or a Hankel matrix is multiplied by a related companion matrix. We present a new condition, generalizing known results, for a Toeplitz or a Hankel matrix to be the transforming matrix for a similarity between a pair of companion matrices. A special case of our main result shows that a Toeplitz or a Hankel matrix can be extended using associated companion matrices, preserving the Toeplitz or Hankel structure respectively.


Introduction and notation
Companion matrices occur not only in matrix analysis but also in many scientific fields. For example, a companion matrix C 1 naturally arises as the system matrix when a dynamic system is represented in state space form [9,12]. When a basis of the state vector space is changed a new system matrix appears and, quite often, the new system matrix is also a companion matrix C 2 in different form and the similarity relation between the new and the old system matrices is realized by a nonsingular basis changing matrix P, i.e. PC 1 P −1 = C 2 or PC 1 = C 2 P.
Here we will call the basis changing matrix P the transformation matrix. In the literature on dynamic systems the transformation matrices are constructed in order to have a better understanding of the state vectors from a certain aspect, and hence they are verified case by case for the similarity between the pair of companion matrices. A question arises naturally: If C 1 and C 2 are a given pair of similar companion matrices, what is in common among those transformation matrices P satisfying (1)? The most obvious approach is to solve equation (1) directly for all solutions P. Although (1) is not hard to solve it is still not clear, after finding the solutions, what is in common among these P's. In this note we give a necessary and sufficient condition for P to satisfy (1) for certain pairs of similar companion matrices C 1 and C 2 . This condition is not deep but it is clear and constructive. We believe that it is worthwhile to add this little piece of information into the existing rich theory in the literature of matrix analysis. As a byproduct of this condition we have found a pattern in the extension of Toeplitz or Hankel matrices by powers of associated companion matrices, preserving the Toeplitz or Hankel structure respectively. This relation has been actually used implicitly in [2] and [4] where the determinant of an extended Toeplitz matrix is computed via powers of determinants of certain companion matrices. Finally, as an application, we give an example linking the state of a dynamic system after any number of iterations to the initial state together with knowledge of the input up to the current time, using just one matrix being the extension of the system matrix by powers of companion matrices.
We assume always that u 1 , u n+1 , v 1 and v n+1 are nonzero, and that u(λ) and v(λ) are co-prime. The "top", "bottom", "left" and "right" companion matrices of the polynomial u(λ) (or the vector u) are defined as When their dependence on u is clear from context we will simply write C t , C b , C l and Cr. The companion matrices of v(λ) are defined in the same way. Under our assumptions on u and v, all the companion matrices defined above are nonsingular. Let J be the flipping matrix For a vector u we denote by u J the vector Ju, and the corresponding polynomial u J (λ) is defined by u J (λ) = u n+1 + un λ + · · · + u 2 λ n−1 + u 1 λ n . For a matrix A we denote by A J the flipping of A about its secondary diagonal, so A J = JA T J. Hankel matrices are symmetric in the usual sense but Toeplitz matrices A are persymmetric, that is, symmetric about their secondary diagonal We also define the companion matrices of u J and denote them by C t (u J ), C b (u J ), C l (u J ) and Cr(u J ). When their dependence on u J is clear from context we will simply write these matrices as C t , C b , C l and Cr. Define the following triangular Toeplitz matrices using the components of u and v : Analogously we define V+ and V− in terms of the components of v.
of the vectors u, v (or the polynomials u(λ), v(λ)) are the n × n matrices with the generating polynomials and n ∑︁ i,j=1 respectively. The Gohberg-Semencul formulae [5,6] imply that the Toeplitz Bezoutian matrix generated by u and v is and the Hankel Bezoutian matrix generated by u and v is It is known [10] that if u(λ) and v(λ) are co-prime then B T and B H are both nonsingular and that B −1 T is Toeplitz and B −1 H is Hankel.

Similarity of companion matrices
We first list some obvious relations among the companion matrices defined in Section 1. Properties: 1. Inversion: 2. Flipping: 3. Transposition: All these properties can be easily verified. Property 1 can also be found in [3].

Similarity
In this section we derive a necessary and sufficient condition for a nonsingular matrix P to satisfy equation (1). For convenience we define a simple operation on square Toeplitz or Hankel matrices. For an invertible Toeplitz matrix the (n − 1) × (n + 1) Toeplitz matrix ∂T, introduced in [7], is obtained by adding one column to the right preserving the Toeplitz structure and then deleting the first row: Similarly, for an invertible Hankel matrix H = TJ, the (n − 1) × (n + 1) Hankel matrix ∂H is obtained by adding one column to the right preserving the Hankel structure and then deleting the last row: a 0 a 1 · · · a n−1 In the following, given a matrix A, we will use A [i:j,k:l] to denote the sub-matrix of A formed by selecting all rows from the ith row to the jth row and all columns from the kth column to the lth column.
Theorem 2.1. The following three statements are equivalent.

T is an invertible matrix satisfying
In such a case C t T, TCr, C b T and TC l are all Toeplitz matrices.
Proof. We first prove statement 2 implies statement 1.
Putting this into the equation above gives ∂Tu = 0, so u ∈ Ker{∂T}. Now we prove statement 1 implies statement 2. It is easy to see that where β is a column given by Since u ∈ Ker{∂T} we have This implies that As a consequence we have that both TCr and C t T are Toeplitz and The proof for the equivalence of statement 1 and statement 3 is similar.
To prove statement 3 implies statement 1 we begin by assuming A = TC l , B = C b T and A = B as before. The structure of C l and C b implies that A [1:n−1,2:n] = T [1:n−1,1:n−1] and B [1:n−1,2:n] = T [2:n,2:n] . Then from A = B we obtain T [1:n−1,1:n−1] = T [2:n,2:n] which means that T is Toeplitz. By equating the first columns of A and B we get Deleting the last row on both sides we obtain (∂T)u = 0. Finally we prove statement 1 implies statement 3. The structure of C l gives where γ is a column given by Since u ∈ Ker{∂T} we have This implies that [︁ 0 a n−1 · · · a 1 a 0 ]︁ u, that is, γ T J is the last row of TC l . From this, by equation (16) and the assumption that T is Toeplitz, ]︁ is Toeplitz and hence TC l = (TC l ) J . On the other hand, since T is Toeplitz, by equation (8) we have As a consequence we have Proof. Let T = HJ. Then T is a non-singular Toeplitz matrix. Obviously u J ∈ Ker{∂H} is equivalent to u ∈ Ker{∂T}. Now we prove that TCr = C t T is equivalent to C t H = HC l . Since H is Hankel we then have By taking the flipping operation on both sides of TCr = C t T we obtain C J r T J = T J C J t which is, by (8) and Equation (19), C t HJ = HC T t J. Then multiplying both sides by J gives By now we can see that Theorem 2.1 and Corollary 2.2 are basically the same thing.
In the remainder of this section we give two examples. They provide insight into the use of similarity transformations in dynamic systems. Example 1. Our condition is constructive, so we can easily build a transformation matrix in a particular pattern that we want without having to solve the matrix equation We can see immediately that this T is actually U −1 + and it confirms the following situation in the field of dynamic system. In [9] the canonical observer form of a dynamic system has, in our notation, system matrix Ao = C l (u), and the canonical observability form of the same dynamic system has system matrix A ob = C b (u). The transformation matrix determined by the dynamics in [9] is U+ such that Ao U+ = U+A ob . In terms of our Theorem 2.1, this is exactly TC l = C b T with T = U −1 + . Example 2. In this example we apply our condition to another example from [9] where the transformation matrix is constructed in a quite complicated way from the dynamics. Consider a linear dynamic system represented in the state space formẋ where x(t) is the state vector, A is the system matrix, B is the input column matrix, C is the output row matrix, w is a scalar input and y is the scalar output. The state space representation is in canonical controller form will transform Ao into Ac by way of Q −1 Ao Q = Ac, that is It is also shown in [9] that Q = −JB T where B T is the Toeplitz Bezoutian Bez T (u, v) where v = (bn , . . . , b 1 , 0) T .
Our condition provides insight into this complicated situation. For this we need some results from Corollary 2.3, 2.10, Theorem 4.2 and 4.5 of [6]. We merely summarize the relevant information in the following Theorem. Bezoutian matrices B T (a, b) and

Theorem 2.3. A necessary and sufficient condition for two non-zero Toeplitz
for some matrix φ with det φ = 1. (a, b) is just a scalar multiple of T −1 .

If T is an invertible Toeplitz matrix and {a, b} is a basis for the kernel of ∂T, then B T
It is well known that T is Toeplitz, (see [6]), and hence which is (20).

Extension
A Toeplitz (or Hankel) matrix can be extended to any size in a Toeplitz (or Hankel) way, by adding more diagonal bands to the existing bands. Theoretical aspects of such an extension, such as the minimum rank of the extension, have been studied in the literature (see [1] and the references therein). What we are concerned with here is a specific way of extending the matrix by multiplication by some associated companion matrices. We hope this extension might have more applications than the ones we will demonstrate at the end of this paper. We will use the similarity relations among companion matrices that have been developed earlier.
Assume A is an n × n matrix. The role that C t plays in the product C t A is to keep the first n − 1 rows of A as the last n − 1 rows of C t A, and to add one new row on the top. The new row added is a linear combination of rows of A. Similarly, the first n − 1 rows of C b A are the last n − 1 rows of A and the last row of C b A is a linear combination of rows of A. This enables us to extend the matrix A in the upward and downward directions as follows. Starting from A, for integers k ≥ l we define T[A : k, l] to be the (n + k − l) × n matrix where γ i (i = 1, 2, . . . , k − l) is the first row of C l+i t A. In similar fashion the effect of post-multiplying a matrix by Cr or C l can be considered. We can extend a matrix in the right and left directions by adding the last column of ACr to the right or adding the first column of AC l to the left. Starting from T[A : k, l], for integers s ≥ t we define where β i (i = 1, 2, . . . , s − t) is the last column of T[A : k, l]C t+i r . We call this a Toeplitz extension because we will prove that, under certain conditions, such an extension preserves the Toeplitz structure if the starting matrix A is Toeplitz. We call A a generator in such an extension. To generate the same matrix T[A : k, l; s, t] we can use any n × n matrix of the form C i t AC j r where i and j are integers. It is easy to see that If we use Cr and C l instead of Cr and C l in the above extension, we will obtain a different extended matrix We notice that, for any square matrix A, An obvious property of these extensions is given in the following Proposition. , where e i is the ith column of the (n + r) × r Toeplitz matrix whose first column is [︁ u 1 · · · u n+1 0 · · · 0 ]︁ T and last column is [︁ 0 · · · 0 u 1 · · · u n+1 ]︁ T . In particular,  , where e J i is the ith column of the (n + r) × r Hankel matrix whose first column is [︁ 0 · · · 0 u n+1 · · · u 1 ]︁ T and last column is [︁ u n+1 · · · u 1 0 · · · 0 ]︁ T .
Applying Lemma 3.3 successively gives that u belongs to the kernel of ∂(C i t TC s r ) for each s = 0, 1, . . . , j. Finally by Theorem 2.1 we conclude that C i t TC j+1 r is Toeplitz. Therefore we have completed the induction for the upward and rightward directions.
In the downward and leftward directions we have negative indices i < 0 and j < 0 in C i t TC j r . By repeating the same induction argument for the negative indices from i to i − 1 and from j to j − 1, we have the Toeplitz extension in the downward and leftward directions. Due to the Toeplitz structure of such an extension we only need to prove these features for the cases when k, l, s and t are integer multiples of n and the general case is just an arbitrary truncation of these special cases. We only demonstrate the proof in the case of k = s = t = n and l = 0. We write It is easy to see that because S 1 = U −1 + and, by Proposition 3.1, S 2 U+ + S 1 U− = 0. We now show that To see this we apply Proposition 3.1 to T[U −1 + : n, 0; n, n]: The first term on the right hand side is equal to I and hence the second term on the right hand side is equal to −I.

Applications
Here we give an example of application of the above extension. This is a problem studied in [8] and we briefly describe it as follows. For an arbitrary sequence y = (y k ) ∞ k=1 its λ-transform (generating function) is defined to beŷ(λ) := ∑︀ ∞ k=1 y k λ k−1 . Consider a linear, time-invariant, causal, discrete-time dynamic system with transfer function descriptionŷ(λ) = −(v(λ)/u(λ))ŵ(λ), where w = (w k ) ∞ k=1 is the input, y = (y k ) ∞ k=1 is the output, and the numerator and the denominator of the transfer function −v(λ)/u(λ) satisfy all the assumptions stated in Section 1, that is, u(λ) = u 1 + u 2 λ +· · ·+ u n+1 λ n and v(λ) = v 1 + v 2 λ +· · ·+ v n+1 λ n with nonzero u 1 , u n+1 , v 1 , v n+1 as well as that u(λ) and v(λ) are co-prime. We will represent this system in state space form by introducing state vectors first and then write down the rule of evolution of state vectors in terms of the initial state and the input.
When both w and y are sequences in l 1 , the analytic functionsŵ(λ) andŷ(λ) are related by Equating like powers of λ gives Uy + Vw = 0 where w and y are column vectors and It can be shown using functional analysis arguments that such w and y have the form (w, y) = (−Ub, Vb) for some b ∈ l∞.
Writing out this equation in detail yields the difference equations and Now we can introduce naturally the n-dimensional state vector at the time k as the truncation b [k:n+k−1] of b ∈ l∞ and denote it by Then we can put (25) and (26) in the state space form where the system matrix is . and the output matrix C is Note that the output matrix C is actually the first row of B T (u, v) divided by u 1 . The general truncation b [1:n+p]  , and s i is the element at the last column and last row of (C b ) i . It can be shown that (1/u 1 )Fp is the inverse of the p × p lower triangular nonsingular truncation of U, but we skip the proof here. Now we change the basis of the state space by using the transformation matrix B T (u, v), that is, we introduce x ′ (k) = B T x(k). Then, by (21), the state space representation of the system is transformed into another canonical form where the system matrix A ′ = C l , and the input and output matrices become Then the state vector at the time q > 0 can be expressed directly in terms of the initial state x ′ (0) and the input data: x ′ (q) = (A ′ ) q x ′ (0) + T[I : 0, 0; 0, 1 − q]Eq where Eq is the (n + q − 1) × q band matrix A combination of (28) and (30) for all positive integers p and q. As we can see that using our extension we can link the state variable at any time directly to the initial state by one equation without having to express it in an iterative way. All extensions involve powers of companion matrices. The formula derived in [11] can be used to calculate entries of an integer power of a companion matrix directly.