# Lecture 6: The Group Inverse

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1 Lecture 6: The Group Inverse

2 The matrix index Let A C n n, k positive integer. Then R(A k+1 ) R(A k ). The index of A, denoted IndA, is the smallest integer k such that R(A k ) = R(A k+1 ), or equivalently, rank A k = rank A k+1 (1) holds. Then R(A j ) = R(A Ind A ), j > IndA. A matrix A is range Hermitian (or EP matrix) if R(A) = R(A ) Special cases: (a) Nonsingular matrices, (b) normal matrices, in particular (c) A = O. If A is range Hermitian, then Ind A = 1. [ ] The converse is not true, e.g., A = a

3 Index and complementary subspaces Theorem. Let A C n n have index k, and let l be a positive integer. Then R(A l ) and N(A l ) are complementary subspaces if and only if l k. Proof. The theorem is obvious for nonsingular matrices. Let A C n n. Then, for any positive integer l, From dimr(a l ) + dimn(a l ) = rank A l + nullitya l = n. C n = R(A l ) N(A l ) R(A l ) N(A l ) = {0}. it follows that R(A l+1 ) R(A l ) and N(A l ) N(A l+1 ), C n = R(A l ) N(A l ) R(A l ) = dimr(a l+1 ). 3

4 Nilpotent matrices A matrix N is nilpotent if N k = O for some integer k 0. The smallest such k is called the index of nilpotency of N. Let J k (λ) be a k k Jordan block, J k (λ) = λ 1 0. λ λ 1 0 λ Then J k (λ) is nilpotent if λ = 0, indj k (λ) = 1, if λ 0; k, if λ = 0. 4

5 Let A C n n, X = Jordan blocks [x 1 x 2 x k ] AX = XJ k (λ), where J k (λ) = C n k k λ λ 1.. Ax 1 = λx 1, and λ C satisfy λ λ Ax j = λx j + x j 1, j = 2,...,k. C k k, j 1, k : (A λi) j x j = 0, (A λi) j 1 x j = x 1 0, x 1 is an eigenvector of A corresponding to λ x j is a λ vector (principal vector, generalized eigenvector) of A of grade j 5

6 J k (λ) = Powers of the matrix (J k (λ) λi) C k k λ λ λ λ (J k (λ) λi) 2 =, J k (λ) λi = , (J k (λ) λi) k = O. 6

7 The Jordan normal form Theorem. Let A C n n. Then A is similar to a block diagonal matrix J with Jordan blocks on its diagonal, i.e. nonsingular X J k1 (λ 1 ) O O O J X 1 k2 (λ 2 ) O AX = J = (1) O O J kp (λ p ) The matrix J, called the Jordan normal form of A, is unique up to a rearrangement of its blocks. The scalars {λ 1,...,λ p } in (1) are the eigenvalues of A. The set of eigenvalues, or spectrum of A, is denoted Λ(A). For A as above, (A λ 1 I) k 1 (A λ 2 I) k2 (A λ p I) k p = O. (2) 7

8 The polynomial Vanishing polynomials c(z) = (z λ 1 ) k 1 (z λ 2 ) k2 (z λ p ) k p is the characteristic polynomial of A. An eigenvalue λ may be repeated, say c(z) = p(z)(z λ) k 1 (z λ) k1 (z λ) k m, p(λ) 0 The algebraic multiplicity of the eigenvalue λ is then k 1 + k k m and max {k i : i 1, m} is the geometric multiplicity, or index of the eigenvalue λ, denoted ν(λ). The polynomial m(z) = (z λ) ν(λ) λ Λ(A) is the minimal polynomial of A. 8-a

9 Spectral inverses Let A C n n. Then X is an S inverse of A if they share the property that, for every λ C and every vector x, x is a λ vector of A of grade p x is a λ vector of X of grade p where λ = 0 if λ = 0, and otherwise 1/λ. Ex. If A C n n is diagonable, A = PJP 1, J = diag(λ 1, λ 2,, λ n ) then X = PJ P 1, J = diag (λ 1, λ 2,, λ n) is an S inverse of A. It is a {1, 2} inverse, and commutes with A, AX = XA. 9

10 The group inverse Theorem. Let A C n n. Then the system has a solution X iff AXA = A, (1) XAX = X, (2) AX = XA, (5) Ind A = 1 and the solution is unique. It is called the group inverse, or {1, 2, 5} inverse, of A, and is denoted A #. Proof. X A{1, 2} AX = P R(A),N(X), XA = P R(X),N(A) X A{1, 2, 5} AX = P R(A),N(A) C n = R(A) N(A) IndA = 1 10

11 The group inverse (cont d) Theorem. A is range Hermitian if, and only if, A # = A. Proof. A = A (1,2) R(A),N(A), A# = A (1,2) R(A ),N(A ). Theorem (Erdélyi). Let A have index 1 and Jordan form A = PJP 1, Then A # = PJ P 1. Proof. The relations (1) AXA = A, (2) XAX = X, and (5) AX = XA, are similarity invariants. Therefore J # = P 1 A # P and since IndJ = 1, J # = J. 11-a

12 The group inverse (cont d) Theorem (Cline). Let a square matrix A have the FRF A = CR. Then A has group inverse if and only if RC is nonsingular, and A # = C(RC) 2 R. Ex. Let A C n n. Then A has index 1 if and only if the limit exists, in which case lim (λi n + A) 1 A λ 0 lim (λi n + A) 1 A = AA #. λ 0 Proof. A full rank factorization A = CR gives (λi n + A) 1 A = C(λI r + RC) 1 R, etc. 12

13 Properties of the group inverse (a) If A is nonsingular, A # = A 1. (b) A ## = A. (c) A # = A #. (d) A T# = A #T. (e) (A l ) # = (A # ) l for every positive integer l. (f) Let A have index 1 and denote Then, A j := (A # ) j, j = 1, 2,..., A 0 := AA #. A l A m = A l+m, l, m. The powers of A, positive, negative and zero, constitute an Abelian group under matrix multiplication ( group inverse ) 13

14 Lemma on λ vectors Lemma 1. Let x be a λ vector of A with λ 0. Then x R(A l ) where l is an arbitrary positive integer. Proof. Let (A λi) p x = 0 for some positive integer p, or, using the binomial expapsion, ( ) p x = c 1 Ax + c 2 A 2 x + + c p A p x, c i = ( 1) i 1 λ i. (1) i Ax = c 1 A 2 x + c 2 A 3 x + + c p A p+1 x, A 2 x = c 1 A 3 x + c 2 A 4 x + + c p A p+2 x, = (2) A l 1 x = c 1 A l x + c 2 A l+1 x + + c p A p+l 1 x, Successive substitutions of (2) in (1) give x = A l q(a)x, where q is some polynomial. 14

15 Lemma on spectral inverses Lemma 2. Let A be a square matrix and let XA l+1 = A l (1) for some positive integer l. Then every λ vector of A of grade p for λ 0 is a λ 1 vector of X of grade p. Proof by induction on the grade p. p = 1. Let Ax = λx, λ 0. Then A l+1 x = λ l+1 x x = λ l 1 A l+1 x Xx = λ l 1 XA l+1 x = λ 1 x. Assume true for p 1, r. Let x be a λ vector of A of grade r

16 Lemma 2 (cont d) Then, by Lemma 1, x = A l y, for some y. (X λ 1 I)x = (X λ 1 I)A l y = X(A l λ 1 A l+1 )y = X(I λ 1 A)A l y = λ 1 X(A λi)x. By the induction hypothesis, (A λi)x is a λ 1 vector of X of grade r. (X λ 1 I) r (A λi)x = 0, z = (X λ 1 I) r 1 (A λi)x 0 Xz = λ 1 z. (X λ 1 I) r+1 x = λ 1 X(X λ 1 I) r (A λi)x = 0, (X λ 1 I) r x = λ 1 X z = λ 2 z 0. x is a λ 1 vector of X of grade r

17 The group inverse is spectral Recall: Let A C n n. Then X is an S inverse of A if they share the property that, for every λ C and every vector x, x is a λ vector of A of grade p x is a λ vector of X of grade p where λ = 0 if λ = 0, and otherwise 1/λ. Theorem. Let A C n n have index 1. Then A # is the unique S inverse of A in A{1} A{2}. If A is diagonable, A # is the only S inverse of A. Proof. A # is an S inverse of A. Since X = A # satisfies XA l+1 = A l with l = 1, it follows from Lemma 2 that A # satisfies the = part of the definition of S inverse for λ 0. Replacing A by A # establishes the = part for λ 0, since A ## = A. The rest of the proof omitted. 17

18 Application to finite Markov chains A system is observed at times t = 0, 1, 2,.... The system has N states, denoted {1, 2,...,N}, the state at time t is denoted X t The system is a (finite) Markov chain (or chain) if a matrix P = (p ij ) R N N such that Prob {X t+1 = j X t = i} = p ij, i, j 1, N, t = 0, 1,.... The numbers p ij are called the transition probabilities, and the matrix P is called the transition matrix. N p ij = 1, i 1, N, (a) j=1 p ij 0, i, j 1, N. (b) A square matrix P = [p ij ] satisfying (a),(b) is called stochastic. Condition (a) is Pe = e, i.e., 1 Λ(P). 18

19 Let p (n) ij Markov chains (cont d) denote the n step transition probability p (n) ij = Prob {X t+n = j X t = i}, p (0) ij := δ ij. The Chapman Kolmogorov equations p (m+n) ij = N k=1 p (m) ik p(n) kj, m, n Z +. (1) p (n) ij is the (i, j) th element of P n. More terminology: (a) A state i leads to state j, denoted by i j, if p (n) ij > 0 for some n 1. (b) Two states i, j communicate, denoted by i j if each state leads to the other. (c) A set S of states is closed if i, j S = i j 19-a

20 Terminology (cont d) (d) A chain is irreducible if its only closed set is the set of all states 1, N, and is reducible otherwise. (e) A single state forming a closed set is an absorbing state. A reducible chain is absorbing if each of its closed sets consists of a single (necessarily absorbing) state. (f) A chain is regular if for some k, P k is a positive matrix. (g) A state i has period τ if p (n) ii = 0 except when n = τ, 2τ, 3τ,.... The period of i is denoted τ(i). If τ(i) = 1, i is aperiodic. (h) Let f (n) ij reached for 1st time at step n, and let f ij = n=1 µ ij = = probability that starting from i, state j is f (n) ij probability that j eventually reached from i, n=1 nf (n) ij the mean 1st passage time from i to j. 20

21 (h) µ kk := n=1 nf (n) kk Terminology (cont d) is the mean return time of state k. (i) A state i is recurrent if f ii = 1, and is transient otherwise. A chain is recurrent if all its states are recurrent. (j) If i is recurrent and µ ii =, i is called a null state. All states in a closed set are of the same type. (k) A state is ergodic if recurrent and aperiodic, but not a null state. If a chain is irreducible, all its states have the same period. An irreducible aperiodic chain is called ergodic. Theorem (Chung). For any states i, j of an irreducible recurrent chain n 1 lim n n+1 p (k) ij = 1. (1) µ jj k=0 Proof. Both sides of (1) give the expected number of returns to state j in unit time. 21-a

22 Stationary probabilities The probabilities {π 1, π 2,...,π N } are stationary if π k = N i=1 p ik π i, k 1, N, or, in terms of vector of stationary probabilities, P T π = π. π is called a stationary distribution or steady state. In an irreducible recurrent chain, the stationary distribution is simply, π k = 1 µ kk, k 1, N. If a chain is ergodic, the system converges to its stationary distribution from any initial distribution. In terms of P, every row of P n converges to π T. 22

23 Summary for ergodic chains Theorem (Feller). Let P R N N be the transition matrix of an ergodic chain. Then: (a) j, k 1, N the following limit exists, independent of j, lim n p(n) jk = π k > 0 (1) (b) π k = 1 µ kk (c) The numbers {π k : k 1, N} are probabilities, π k 0, N k=1 and are a stationary distribution of the chain: π k = N i=1 π k = 1, (2) p ik π i, k 1, N. (3) The distribution {π k } is uniquely determined by (2) and (3). 23

24 Solution using the group inverse Theorem (Meyer). Let P be the transition matrix of a finite Markov chain, and let Q = I P. Then Q has a group inverse, and: I QQ # = lim n 1 n + 1 n k=0 P k, (1) = lim n (αi + (1 α)p)n, for any 0 < α < 1, (2) and if the chain is regular, or absorbing, I QQ # = lim n P n. (3) For proofs and other details see original work. 24

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