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Lecture 17 Perron-Frobenius Theory - Stanford University

EE363 Winter 2008-09 Lecture 17 Perron-Frobenius Theory Positive and nonnegative matrices and vectors Perron-Frobenius theorems Markov chains Economic growth Population dynamics Max-min and min-max characterization Power control Linear Lyapunov functions Metzler matrices17 1 Positive and nonnegative vectors and matriceswe say a matrix or vector is positive(orelementwise positive) if all its entries are positive nonnegative(orelementwise nonnegative) if all its entries arenonnegativewe use the notationx > y(x y) to meanx yis elementwise positive(nonnegative)warning:ifAandBare square and symmetric,A Bcan mean: A Bis PSD ( ,zTAz zTBzfor allz), or A Belementwise positive ( ,Aij Bijfor alli, j)in this Lecture ,>and mean elementwisePerron- frobenius Theory17 2 Application areasnonnegative matrices arise in many fields, , economics population models graph Theory Markov chains power control in communications Lyapunov analysis of large scale systemsPerron- frobenius Theory17 3 Basic factsifA 0andz 0, then we haveAz 0conversely: if for allz 0, we haveAz 0, then we can concludeA 0in other words, matrix multiplication preserves nonnegativity if and only ifthe matrix is nonnegativeifA >0andz 0,z6= 0, thenAz >0conversely, if wheneverz 0,z6= 0, we haveAz >0, then we canconcludeA >0ifx 0andx6= 0, we refer tod= (1/1Tx)xas itsdistributionornormalized formdi=xi/(Pjxj)gives the fraction of the total ofx, given byxiPerron- frobenius Theory17 4 Regular nonnegative matricessupposeA

nonnegative matrices arise in many fields, e.g., • economics • population models • graph theory • Markov chains • power control in communications • Lyapunov analysis of large scale systems Perron-Frobenius Theory 17–3

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