1 Algebra, definitions1.1 Rings (from Control System Synthesis, Vidyasagar, MIT Press)A ring consists of: a set R, and two operations, addition (+) and multipli-cation (·), each mapping R × R → R. The operations must sa tisfy severalaxiom s.• For every a, b ∈ R, there corresponds an element u+w ∈ R, the additio nof a and b. For all a, b, c ∈ R, it must be that1. a + b = b + a2. (a + b) + c = a + (b + c)There is a unique element θ ∈ R (o r 0R, θR, o r just 0) such that forevery a ∈ R, a + θ = a. Moreover, for every a ∈ R, there i s a uniqueelement labled −a such that a + (−a) = θ.• For every a, b, c ∈ R a · (b · c) = (a · b) · c.• For every a, b, c ∈ R1. a · (b + c) = a · b + a · c2. (a + b) · c = a · c + b · cA ring R i s commutative if a · b = b · a for all a, b ∈ R.A ring R is said to ha v e an ide ntity if there is an element in R, denoted 1,such that for every a ∈ R, 1 · a = a · 1 = a.A ring R i s a domain if a, b ∈ R, a · b = 0 implies either a = 0 or b = 0.If R is a ring with identity, then a ∈ R is called a unit of R if there is anelement b ∈ R such that a · b = b · a = 1. In this case, b is the inverse of a,and denoted by a−1.1.1.1 ExamplesExample: Let R be the set of 2 × 2 matrices with integer entries.19Example: Let R be the set of even integersExample: Let R be the set of integersExample: Let R be the set of polynomia ls in a single indeterminate variable,say s. The coefficients are complex numb ers. We could also restrict thecoeffici ents to be real. This is i mportant, and we denote it as P.Example: Let R be the set of exponentially stable, linear finite-dimensionalsystems (“multiplication” defined as composition).Example: Let R be the set of tra nsfer functio ns of exponentially st able,linear finite-dimensional syst ems. T his is important, and we denote it by S.HenceS :=n(s)d(s): n, d ∈ P, ∂n ≤ ∂d, d(s0) 6= 0∀Re(s0) ≥ 01.2 FieldsA field is a commutative ring R with identity, containing at least two ele-ments, with one additional property: every nonzero element of R is a unit.Usually use F (rather than R) to denote a field.1.3 Matrices with elements in Commutative RingsSuppose K be a commutative ring. Now let H ∈ Kn×n, so that H is a squarematrix whose elements are in K. Recall that• matrix multi plicati on, and matr ix addi tion• determinant of a matrix, and adjoint of a matrixare defined in terms of products (multiplication) and sums ( addition) of theelements of a matrix.It is easy to go back to chapter 2 (probably) in your linear algebra book ,and verify that much of the linear algebra we had stated for determinants20(of real and/or com plex matrices) along the Cramer ’s rule also holds formatrices w ith elements in commutative rings, soA ∈ Kn×n⇒ det(A) ∈ KA ∈ Kn×n⇒ adj(A) ∈ Kn×nA ∈ Kn×m, B ∈ Km×n⇒ AB ∈ Kn×nA ∈ Kn×m, B ∈ Km×n⇒ det(AB) = det(A) det(B)H ∈ Kn×n⇒ Hadj(H) = adj(H)H = det(H)In.These lead to:Theorem: Given H ∈ Kn×n. There is a m atrix W ∈ Kn×nsuch thatW H = HW = Inif and only if det(H) ∈ UK.⇒ 1 = det(In) = det(W H) = det(W ) det(H). Since det(W ) ∈ K it followsthat det(H) ∈ UK.⇐ Let D ∈ K be the mult iplicat ive inverse of det(H). U sing Cr amer’s rule(and that D is a scalar ) we have H (Dadj(H)) = (Dadj(H)) H =