A standard form in (some) free fields: How to construct minimal linear representations

Definition 1.1

(Inner Rank, Full Matrix, Hollow Matrix [5,9]) Given a matrix A ∈ K 〈 X 〉 n × n , the inner rank of A is the smallest number m ∈ ℕ such that there exists a factorization A = T U with T ∈ K 〈 X 〉 n × m and U ∈ K 〈 X 〉 m × n . The matrix A is called full if m = n , non-full otherwise. It is called hollow if it contains a zero submatrix of size k × l with k + l > n .

Definition 1.2

(Associated and Stably Associated Matrices [10]) Two matrices A and B over K 〈 X 〉 (of the same size) are called associated over a subring R ⊆ K 〈 X 〉 if there exist invertible matrices P , Q over R such that A = P B Q . A and B (not necessarily of the same size) are called stably associated if A ⊕ I p and B ⊕ I q are associated for some unit matrices I p and I q . Here by C ⊕ D we denote the diagonal sum C . . D .

Lemma 1.3

[10, Corollary 6.3.6] A linear square matrix over K 〈 X 〉 which is not full is associated over K with a linear hollow matrix.

Definition 1.4

(Linear Representations, Dimension, Rank [6,9]) Let f ∈ F . A linear representation of f is a triple π f = ( u , A , v ) with u ∈ K 1 × n , full A = A 0 ⊗ 1 + A 1 ⊗ x 1 + ⋯ + A d ⊗ x d , that is, A is invertible over F , ∀ ℓ A ℓ ∈ K n × n , v ∈ K n × 1 and f = u A − 1 v . The dimension of π f is dim ( u , A , v ) = n . It is called minimal if A has the smallest possible dimension among all linear representations of f. The “empty” representation π = ( , , ) is the minimal one of 0 ∈ F with dim π = 0 . Let f ∈ F and π be a minimal linear representation of f. Then the rank of f is defined as rank f = dim π .

Remark

Cohn and Reutenauer defined linear representations slightly more generally, namely, f = c + u A − 1 v with possibly non-zero c ∈ K and call it pure when c = 0 . Two linear representations are called equivalent if they represent the same element [9]. Two (pure) linear representations ( u , A , v ) and ( u ˜ , A ˜ , v ˜ ) of dimension n are called isomorphic if there exist invertible matrices P , Q ∈ K n × n such that u = u ˜ Q , A = P A ˜ Q and v = P v ˜ [9].

Theorem 1.5

[9, Theorem 1.4] If π ′ = ( u ′ , A ′ , v ′ ) and π ″ = ( u ″ , A ″ , v ″ ) are equivalent (pure) linear representations, of which the first is minimal, then the second is isomorphic to a representation π = ( u , A , v ) which has the block decomposition

Remark

In principle, given π ″ of dimension n, one can look for invertible matrices P , Q such that P π ″ Q = ( u ″ Q , P A ″ Q , P v ″ ) has the form of π to minimize π ″ . However, for an alphabet with d letters and a lower left block of zeros of size k × ( n − k ) we would get ( d + 1 ) k ( n − k ) + k polynomial equations with at most quadratic terms and two equations of degree n (to ensure invertibility of the transformation matrices) in 2 n 2 commuting unknowns. This is already rather challenging for n = 5 . The goal is therefore to use linear techniques as far as possible.

Definition 1.6

(Left and Right Families [6]) Let π = ( u , A , v ) be a linear representation of f ∈ F of dimension n. The families ( s 1 , s 2 , … , s n ) ⊆ F with s i = ( A − 1 v ) i and ( t 1 , t 2 , … , t n ) ⊆ F with t j = ( u A − 1 ) j are called left family and right family, respectively. L ( π ) = span { s 1 , s 2 , … , s n } and R ( π ) = span { t 1 , t 2 , … , t n } denote their linear spans (over K ).

Proposition 1.7

[6, Proposition 4.7] A representation π = ( u , A , v ) of an element f ∈ F is minimal if and only if both the left family and the right family are K -linearly independent. In this case, L ( π ) and R ( π ) depend only on f.

Definition 1.8

(Element Types [13]) An element f ∈ F is called of type ( 1 , ∗ ) (respectively, ( 0 , ∗ ) ) if 1 ∈ R ( f ) , that is, 1 ∈ R ( π ) for some minimal linear representation π of f (respectively, 1 ∉ R ( f ) ). It is called of type ( ∗ , 1 ) (respectively, ( ∗ , 0 ) ) if 1 ∈ L ( f ) (respectively, 1 ∉ L ( f ) ). Both subtypes can be combined.

Remark

The following definition is a special case of Cohn’s more general admissible systems [2, Section 7] and the slightly more general linear representations [6].

Definition 1.9

(Admissible Linear Systems [11]) A linear representation A = ( u , A , v ) of f ∈ F is called admissible linear system (ALS) for f if u = e 1 = [ 1 , 0 , … , 0 ] . The element f is then the first component of the (unique) solution vector s = A − 1 v . An ALS is also written as A s = v , or if v = [ 0 , … , 0 , λ ] as ( 1 , A , λ ) . Given a linear representation A = ( u , A , v ) of dimension n of f ∈ F and invertible matrices P , Q ∈ K n × n , the transformed P A Q = ( u Q , P A Q , P v ) is again a linear representation (of f). If A is an ALS, the transformation ( P , Q ) is called admissible if the first row of Q is e 1 = [ 1 , 0 , … , 0 ] .

Remark

The left family ( A − 1 v ) i (respectively, the right family ( u A − 1 ) j ) and the solution vector s of A s = v (respectively, t of u = t A ) are used synonymously.

Remark 1.10

While it was intended in [12] to derive a “better” standard form including knowledge about the factorization it turned out later that this is not that easy in the general case because of the necessity to distinguish several cases in the multiplication [13, Section 5]. The term “pre-standard ALS” (for polynomials) is now replaced by polynomial ALS (which is just a special form of a refined ALS, Section 3). And a minimal polynomial ALS is already in a standard form. Particularly to avoid confusion with special transformation matrices for the factorization, everything is put into a uniform context in [22]. For an overview see also [7, Figure 1]. There are only some minor changes in the formulation of results and proofs necessary, for example, to construct the product in [12, Lemma 39] using λ 2 λ A 1 and λ λ 2 A 2 .

Definition 1.11

(Polynomial ALS and Transformation [12]) An ALS A = ( u , A , v ) of dimension n with system matrix A = ( a i j ) for a non-zero polynomial 0 ≠ p ∈ K 〈 X 〉 is called polynomial, if

1. v = [ 0 , … , 0 , λ ] T for some λ ∈ K and

2. a i i = 1 for i = 1 , 2 , … , n and a i j = 0 for i > j , that is, A is upper triangular.

If additionally α 1 , n = α 2 , n = … = α n − 1 , n = 0 , then ( P , Q ) is called polynomial factorization transformation.

Definition 1.12

Let M = M 1 ⊗ x 1 + ⋯ + M d ⊗ x d with M i ∈ K n × n for some n ∈ ℕ . An element in F is called regular if it has a linear representation ( u , A , v ) with A = I − M , that is, A 0 = I in Definition 1.4, or equivalently, if A 0 is regular (invertible).

Proposition 2.1

(Minimal Monomial [11, Proposition 4.1]) Let k ∈ ℕ and f = x i 1 x i 2 … x i k be a monomial in K 〈 X 〉 ⊆ K ( 〈 X 〉 ) . Then

is a minimal polynomial ALS of dimension dim ( A ) = k + 1 .

Proposition 2.2

(Rational Operations [9]) Let 0 ≠ f , g ∈ F be given by the admissible linear systems A f = ( u f , A f , v f ) and A g = ( u g , A g , v g ) , respectively, and let 0 ≠ μ ∈ K . Then admissible linear systems for the rational operations can be obtained as follows:

The scalar multiplication μ f is given by

The sum f + g is given by

The product fg is given by

And the inverse f − 1 is given by

Lemma 2.3

[12, Lemma 25]. Let A = ( u , A , v ) be an ALS of dimension n ≥ 1 with K - linearly independent left family s = A − 1 v and B = B 0 ⊗ 1 + B 1 ⊗ x 1 + ⋯ + B d ⊗ x d with B ℓ ∈ K m × n , such that B s = 0 . Then there exists a (unique) T ∈ K m × n such that B = T A .

Lemma 2.4

(For Type (∗, 1) [11, Lemma 4.11]) Let A = ( u , A , v ) be a minimal ALS with dim A = n ≥ 2 and 1 ∈ L ( A ) . Then there exists an admissible transformation ( P , Q ) such that the last row of PAQ is [ 0 , … , 0 , 1 ] and P v = [ 0 , … , 0 , λ ] ⊤ for some λ ∈ K .

Lemma 2.5

(For Type (1, ∗) [11, Lemma 4.12]) Let A = ( u , A , v ) be a minimal ALS with dim A = n ≥ 2 and 1 ∈ R ( A ) . Then there exists an admissible transformation ( P , Q ) such that the first column of PAQ is [ 1 , 0 , … , 0 ] ⊤ and P v = [ 0 , … , 0 , λ ] ⊤ for some λ ∈ K .

Remark

If g is of type ( ∗ , 1 ) , then, by Lemma 2.4, each minimal ALS for g can be transformed into one with the last row of the form [ 0 , … , 0 , 1 ] . If g is of type ( 1 , ∗ ) , then, by Lemma 2.5, each minimal ALS for g can be transformed into one with the first column of the form [ 1 , 0 , … , 0 ] ⊤ . This can be done by linear techniques, see the remark before [11, Theorem 4.13].

Remark

Since p ∈ K 〈 X 〉 is of type ( 1 , 1 ) , both constructions can be used for the minimal polynomial multiplication (Proposition 2.10). The alternative proof in [13] relies in particular on Lemma 2.9. One could call the multiplication from Proposition 2.2 type ( ∗ , ∗ ) . For a discussion of minimality of different types of multiplication we refer to [13].

Proposition 2.6

(Multiplication Type (1, ∗) [13, Proposition 3.11]) Let f , g ∈ F \ K be given by the admissible linear systems A f = ( u f , A f , v f ) = ( 1 , A f , λ f ) of dimension n f of the form

and A g = ( u g , A g , v g ) = ( 1 , A g , λ g ) of dimension n g , respectively. Then an ALS for fg of dimension n = n f + n g − 1 is given by

Proposition 2.7

(Multiplication Type (∗, 1) [13, Proposition 3.12]) Let f , g ∈ F \ K be given by the admissible linear systems A f = ( u f , A f , v f ) = ( 1 , A f , λ f ) of dimension n f and A g = ( u g , A g , v g ) = ( 1 , A g , λ g ) of dimension n g of the form

respectively. Then an ALS for fg of dimension n = n f + n g − 1 is given by

Remark

Note that the transformation in the following lemma is not necessarily admissible. However, except for n = 2 (which can be treated by permuting the last two elements in the left family), it can be chosen such that it is admissible. The proof is similar to that of [12, Lemma 2.4].

Lemma 2.8

[13, Lemma 3.15] Let A = ( u , A , v ) be an ALS of dimension n ≥ 2 with v = [ 0 , … , 0 , λ ] ⊤ and K -linearly dependent left family s = A − 1 v . Let m ∈ { 2 , 3 , … , n } be the minimal index such that the left subfamily s ̲ = ( A − 1 v ) i = m n is K -linearly independent. Let A = ( a i j ) and assume that a i i = 1 for 1 ≤ i ≤ m and a i j = 0 for j < i ≤ m (upper triangular m × m block) and a i j = 0 for j ≤ m < i (lower left zero block of size ( n − m ) × m ). Then there exists matrices T , U ∈ K 1 × ( n + 1 − m ) such that

Lemma 2.9

[13, Lemma 3.16]. Let p ∈ K 〈 X 〉 \ K and g ∈ F \ K be given by the minimal admissible linear systems A p = ( u p , A p , v p ) and A g = ( u g , A g , v g ) of dimensions n p and n g , respectively, with 1 ∈ R ( g ) . Then the left family of the admissible linear systems A = ( u , A , v ) for pg of dimension n = n p + n g − 1 from Proposition 2.7 is K -linearly independent.

Proposition 2.10

(Minimal Polynomial Multiplication [12, Proposition 26]) Let 0 ≠ p , q ∈ K 〈 X 〉 be given by the minimal polynomial admissible linear systems A p = ( 1 , A p , λ p ) and A q = ( 1 , A q , λ q ) of dimension n p , n q ≥ 2 , respectively. Then the ALS A from Proposition 2.6 for pq is minimal of dimension n = n p + n q − 1 .

Theorem 2.11

(Minimal Inverse [11, Theorem 4.13]) Let f ∈ F \ K be given by the minimal admissible linear system A = ( u , A , v ) of dimension n. Then a minimal ALS for f − 1 is given in the following way:

f of type ( 1 , 1 ) yields f − 1 of type ( 0 , 0 ) with dim ( A ′ ) = n − 1 :

f of type ( 1 , 0 ) yields f − 1 of type ( 1 , 0 ) with dim ( A ′ ) = n :

f of type ( 0 , 1 ) yields f − 1 of type ( 0 , 1 ) with dim ( A ′ ) = n :

f of type ( 0 , 0 ) yields f − 1 of type ( 1 , 1 ) with dim ( A ′ ) = n + 1 :

(Recall that the permutation matrix Σ reverses the order of rows/columns.)

Corollary 2.12

Let p ∈ K 〈 X 〉 with rank p = n ≥ 2 . Then rank ( p − 1 ) = n − 1 .

Corollary 2.13

Let 0 ≠ f ∈ F . Then f ∈ K if and only if rank ( f ) = rank ( f − 1 ) = 1 .

Remark

Note that n ≥ 2 for type ( 1 , 1 ) , ( 1 , 0 ) and ( 0 , 1 ) . The block B is always square of size n − 2 . For n = 2 , the system matrix of A is

1 b . 1 for type ( 1 , 1 ) ,   1 b . c for type ( 1 , 0 )  and   a b . 1 for type ( 0 , 1 ) .

Definition 3.1

(Pivot Blocks, Block Transformation) Let A = ( u , A , v ) be an admissible linear system and denote A = ( A i j ) i , j = 1 m the block decomposition (with square diagonal blocks A i i ) with maximal m such that A i j = 0 for i > j . The diagonal blocks A i i are called pivot blocks, the number m is denoted by # pb ( A ) = # pb ( A ) . The dimension (or size) of a pivot block A i i for i ∈ { 1 , 2 , … , m } is dim i ( A ) . For i < 1 or i > m let dim i ( A ) = 0 . A (admissible) transformation ( P , Q ) is called (admissible) block transformation (for A ) if P i j = Q i j = 0 for i > j (with respect to the block structure of A ).

Notation

Let A = ( u , A , v ) be an ALS with m pivot blocks of size n i = dim i ( A ) . We denote by n i : j = dim i : j ( A ) = n i + n i + 1 + ⋯ + n j the sum of the sizes of the pivot blocks A i i to A j j (with the convention n i : j = 0 for j < i ). For a given system, the identity matrix of size n i : j is denoted by I i : j . If ( P , Q ) is an admissible transformation for A , then ( P A Q ) i j denotes the (to the block decomposition of A corresponding) block ( i , j ) of size n i × n j in PAQ, ( P v ) i ̲ that of size n i × 1 in Pv and ( u Q ) j ̲ that of size 1 × n j in uQ.

Notation

Components in the left family s = A − 1 v are (as usual) denoted by s i . The jth component for 1 ≤ j ≤ dim i ( A ) of the ith block for 1 ≤ i ≤ # pb ( A ) is denoted by s i ̲ ( j ) . A subfamily of s with respect to the pivot block k is denoted by s k ̲ , s i : j ̲ = ( s i ̲ , s i + 1 ̲ , … , s j ̲ ) . Analogous is used for the right family t = u A − 1 .

Notation

A “grouping” of pivot blocks { i , i + 1 , … , j } of the system matrix is denoted by A i : j , i : j . If it is clear from the context where the block ends (respectively, starts), we write A i : , i : (respectively, A : j , : j ), in particular with respect to a given pivot block. For example, A 1 : , 1 : , A k , k and A : m , : m .

Definition 3.2

(Admissible Pivot Block Transformation) Let A = ( u , A , v ) be an ALS with m = # pb ( A ) pivot blocks of size n i = dim i ( A ) . An admissible transformation ( P , Q ) of the form ( I 1 : k − 1 ⊕ T ¯ ⊕ I k + 1 : m , I 1 : k − 1 ⊕ U ¯ ⊕ I k + 1 : m ) with T ¯ , U ¯ ∈ K n k × n k is called admissible for pivot block k or (admissible) k-th pivot block transformation.

Definition 3.3

(Refined Pivot Block and Refined ALS) Let A = ( u , A , v ) be an ALS with m = # pb ( A ) pivot blocks of size n i = dim i ( A ) . A pivot block A k k (for 1 ≤ k ≤ m ) is called refined if there does not exist an admissible pivot block transformation ( P , Q ) k such that ( P A Q ) k k has a lower left block of zeros of size i × ( n k − i ) for an i ∈ { 1 , 2 , … , n k − 1 } . The admissible linear system A is called refined if all pivot blocks are refined.

Remarks

That the “form” of a refined ALS is not unique, it will be illustrated in the following example. Moreover, a refined ALS is not necessarily refined over K ¯ : Let

Adding 2 -times row 1 to row 2 and subtracting 2 -times column 2 of column 1 yields

See also [12, Example 37].

Example 3.4

Let f = ( y − 1 − x ) − 1 be given by the minimal ALS

For f + 3 z , we have the (minimal) ALS

Since 1 ∈ R ( A ) and A is constructed by the addition in Proposition 2.2, it can easily be transformed – in a controlled way – into another ALS with refined pivot block structure. First, we add column 3 to column 1,

and then we exchange rows 1 and 3:

Definition 3.5

(Block Decomposition of an ALS, Block Row and Column Transformation) Let A = ( u , A , v ) be an ALS of dimension n with m = # pb ( A ) ≥ 2 pivot blocks of size n i = dim i ( A ) . For 1 ≤ k ≤ m , the block decomposition with respect to the pivot block k is the system

with (square) diagonal blocks A 1 : , 1 : , A k , k and A : m , : m of size n 1 : k − 1 , n k respectively n k + 1 : m . ( k ̲ is used here to emphasize that k is a block index.) By A [ − k ̲ ] we denote the ALS A [ k ̲ ] without block row/column k of dimension n − n k (not necessarily equivalent to A [ k ̲ ] ):

An admissible transformation ( P , Q ) k ̲ = ( P ( T ¯ , T ) , Q ( U ¯ , U ) ) k ̲ of the form

is called kth block row transformation for A [ k ̲ ] , one of the form ( P ( T ¯ , T ) , Q ( U ¯ , U ) ) k ̲ ,

is called kth block column transformation for A [ k ̲ ] . For T ¯ = U ¯ = I n k we write also P ( T ) , respectively, Q ( U ) and call the block transformation particular.

Definition 3.8

(Standard Admissible Linear System) A minimal and refined ALS A = ( u , A , v ) = ( 1 , A , λ ) , that is, v = [ 0 , … , 0 , λ ] , is called standard.

Remark

For a polynomial p given by a standard ALS A (of dimension n ≥ 2 ), the minimal inverse of A (of dimension n − 1 ) is refined if and only if A is obtained by the minimal polynomial multiplication of its irreducible factors q i in p = q 1 q 2 ⋯ q m . For a detailed discussion about the factorization of polynomials (in free associative algebras) we refer to [12]. One of the simplest non-trivial examples is

With respect to the general factorization theory it is open to show that the free field is a “similarity unique factorization domain” [13].

Example 3.9

(Factorization versus Refinement) Later, in Example 5.1 (Hua’s identity), we need to refine a pivot block. Although the necessary transformations are obvious, the procedure should be illustrated in a systematic way. But first we have a look on how this 2 × 2 block appears, namely, by inverting the element given by the ALS

If we exchange columns 2 and 3, it is immediate that this is the “product” of the admissible linear systems

Applying the minimal inverse on