13 - Digraphs

13 - Digraphs - 1 Math 1b Practical February 22, 2011...

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1 Math 1b Practical — February 22, 2011 Digraphs, nonnegative matrices, Google page ranking — Nonnegative matrices — Theorem 1. Let A be a square matrix of nonnegative real numbers. Then A has an eigenvector e all of whose entries are nonnegative. Proof: Omitted. ut There may be several linearly independent nonnegative eigenvectors. For example, the identity matrix I has many nonnegative eigenvectors (all corresponding to eigenvalue 1). A nonnegative diagonal matrix with distinct diagonal entries has nonnegative eigenvectors corresponding to diferent eigenvalues. Here is a stronger version o± Theorem 1, and also a strengthening when the nonnegative matrix is irreducible in a sense to be de²ned later. Theorem 2. Let A be a square matrix of nonnegative real numbers. Let λ be the max- imum of the absolute values | μ | of the eigenvalues μ of A .T h e n λ is an eigenvalue of A and there are nonnegative eigenvectors coresponding to λ . Also omitted. ut Theorem 3 (Perron-Frobenius). Let A be an irreducible square nonnegative matrix. Then A has a unique nonnegative eigenvector e , up to scalar multiples. This eigenvector e has all positive entries and the corresponding eigenvalue λ has algebraic multiplicity 1 and the property that λ ≥| μ | for every eigenvalue μ of A . Omitted yet again, though we will prove it ±or probability matrices (to be de²ned later) below. The eigenvalue λ in Theorem 2 or 3 may be called the dominant eigenvalue o± the matrix A . We remark that there may be real or complex eigenvalues μ 6 = λ but so that | μ | = λ . Such eigenvalues, o± course, do not correspond to nonnegative eigenvectors, because μ would be negative or complex. For example, ± 01 10 ² has eigenvalues +1 and 1. (In general, the maximum o± the absolute values o± the eigenvalues o± a matrix is called the spectral radius o± the matrix.)
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2 Example 1. The eigenvalues of 0500000030 0000100014 0040003014 1520035000 2020003005 0000004010 0133000005 2100231542 0000220302 0500040043 are 11 . 268 , 5 . 167 , 3 . 556 ± 2 . 351 i, 3 . 269 , 2 . 178 ± 2 . 212 i, 2 . 167 ± 1 . 734 i, 0 . 570 . For this matrix, there are three nonnegative eigenvalues, but only one nonnegative eigen- vector, up to scalar multiples, namely (0 . 148 , 0 . 161 , 0 . 343 , 0 . 329 , 0 . 310 , 0 . 142 , 0 . 328 , 0 . 572 , 0 . 286 , 0 . 305) > . Theorem 4. An eigenvalue λ of a nonnegative matrix A =( a ij ) that corresponds to a nonnegative eigenvector is at least the smallest row sum and at least the smallest column sum of A , and is at most the largest row sum and at most the largest column sum of A . That is, min i n ± j =1 a ij λ max i n ± j =1 a ij and min j n ± i =1 a ij λ max j n ± i =1 a ij .
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This note was uploaded on 11/10/2011 for the course MA 1B taught by Professor Aschbacher during the Winter '08 term at Caltech.

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13 - Digraphs - 1 Math 1b Practical February 22, 2011...

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