2.4 Irreducible Matrices. Reducible is reducible if there is a permutation P such that where A 11 and A 22 are square matrices each of size at least one;

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Presentation transcript:

2.4 Irreducible Matrices

Reducible is reducible if there is a permutation P such that where A 11 and A 22 are square matrices each of size at least one; otherwise A is called irreducible.

1 x 1 matrix: irr or reducible By definition, every 1 x 1 matrix is irreducible. Some authors refer to irreducible if a≠0 reducible if a=0

digraph Let the digraph of A is the digraph with denoted by G(A).

Example for diagraph Let G(A) is 1 2

strongly connected A digraph is called strongly connected if any vertices x,y, there is a directed path from x to y, and vice versa.

Remark Let Given If,then there is a directed walk in G(A) of length l from vertex i to vertex j. If A is nonnegative, then converse also holds

An Equivalent relation on V Define a relation ~ on V by i~j if i=j or i≠j and there is a directed walk from vertex i to vertex j and vice versa. ~ is an equivalent relation.

Strongly Connected Component The strongly connected components are precisely the subgraphs induced by vertices that belong to a equivalent class.

How many strongly connected components are there ? see next page

There are five strongly connected components. final strongly connected component final strongly connected component

Theorem Let The following conditions are equivalent: (a) A is irreducible. (b) There does not exist a nonempty proper subset I of <n> such that (c) The graph G(A) is strongly connected.

Exercise p.1 (a) Show that a square matrix A is reducible if and only if there exists a permutation such that where A 11,A 22 are square matrices each of size at least one.

Exercise p.2 (b) Deduce that if A is reducible, then so is A T

Theorem Let The following conditions are equivalent: (a) A is irreducible. (b) A has no eigenvector which is semipositive but not positive.i.e. every semipositive eigenvector of A is positive (c)

Remark If the degree of minimal polynomial of is m,then

Theorem Let A is irreducible if and only if where m is the degree of minimal polynomial of A.

Exercise Let A is irreducible if and only if for any semipositive vector x, if then x>0

Theorem p.1 (Perron’s Thm) (b) (c) (a)

(f) (g) (e) (d) A has no nonnegative eigenvector other than (multiples of) u.

Theorem (Perron-Frobenius Thm), thenIf and

Corollary Let If A is irreducible, then the conclusions (a),(b),(c),(d) and (f) of Perron Thm all hold.

Exercisse Let Prove that if u is positive and y is nonzero then there is a unique real scalar c such that u+cy is semipositive but not positive.

Remark 2.4.9, then x must be If A is nonnegative irreducible matrix and if x is nonzero nonnegative vector such that the Perron vector of A.

Exercise (n,1) is irreducible. Show that the nxn permutation matrix with 1’s in positions (1,2), (2,3), …, (n-1,n) and

Exercise (a) (a) If A is irreducdible, then A T is irreducdible In below A and B denote arbitrary nxn nonnegative matrices. Prove or disprove the following statements:

Exercise (b) (b) If A is irreducible and p is a positive integer, then A p is irreducible.

Exercise (c) (c) If A p is irreducible for some positive integer, then A is irreducible.

Exercise (d) (c) If A and B are irreducible, then A+B is irreducible.

Exercise (e) (e) If A and B are irreducible, then AB is irreducible.

Exercise (f) (f) If all eigenvalues of A are 0, then A is reducible.

Exercise (g) The matrix is reducible.

G(A) is strongly connected, then A is irreducible 1 2 3

Exercise (h) The matrix is reducible.

G(A) is not strongly connected, then A is reducible

Exercise (i) The matrix is reducible.

G(A) is strongly connected, then A is irreducible

Exercisse (j) A is irreducible if and only if is irreducible.

Exercisse (k) If AB is irreducible, then BA is irreducible

Exercise Let A be an nxn irreducible nonnegative matrix. Prove that if then all row sums of A are equal. Give an example to show that the result no longer holds if the irreducibility assumption is removed.