The Perron-Frobenius  Theorem Consider a non-zero linearoperator B on Rn thatsends the non-negativeorthant into itself.

The Perron-Frobenius  Theorem Consider a non-zero linearoperator B on Rn thatsends the non-negativeorthant into itself.

Let S be the simplex whosecorners are the standardbasis vectors.

S

The Perron-Frobenius  Theorem Consider a non-zero linearoperator B on Rn thatsends the non-negativeorthant into itself.

Let S be the simplex whosecorners are the standardbasis vectors.

Every ray from the origininto the non-negativeorthant hits S at exactly onepoint, so we can identify Swith the space of such rays.

S

The Perron-Frobenius  Theorem Consider a non-zero linearoperator B on Rn thatsends the non-negativeorthant into itself.

Let S be the simplex whosecorners are the standardbasis vectors.

Every ray from the origininto the non-negativeorthant hits S at exactly onepoint, so we can identify Swith the space of such rays.

S

The Perron-Frobenius  Theorem The map B sends S to atriangle floating in thenon-negative orthant.

S

BS

The Perron-Frobenius  Theorem The map B sends S to atriangle floating in thenon-negative orthant.

Project that triange backonto S along rays throughthe origin.

This gives a map b from Sto itself, which describeswhat B does to rays in thenon-negative orthant.

By construction, b is aprojective transformation.

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bS

BS

The Perron-Frobenius  Theorem In particular, b iscontinuous. Since S is aconvex, compact subset ofa Euclidean space, theBrouwer fixed-pointtheorem guarantees thatb has a fixed point.

S

bS

BS

The Perron-Frobenius  Theorem In particular, b iscontinuous. Since S is aconvex, compact subset ofa Euclidean space, theBrouwer fixed-pointtheorem guarantees thatb has a fixed point.

S

bS

BS

The Perron-Frobenius  Theorem In particular, b iscontinuous. Since S is aconvex, compact subset ofa Euclidean space, theBrouwer fixed-pointtheorem guarantees thatb has a fixed point.

S

bS

BS

The Perron-Frobenius  Theorem In particular, b iscontinuous. Since S is aconvex, compact subset ofa Euclidean space, theBrouwer fixed-pointtheorem guarantees thatb has a fixed point.

Every fixed point of bcorresponds to aneigenray of B.

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bS

BS

The Perron-Frobenius  Theorem In general, b can havemany fixed points.

If some power of B mapsthe non-negative orthantinto the positive orthant,however, then b hasjust one fixed point.

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The Perron-Frobenius  Theorem In general, b can havemany fixed points.

If some power of B mapsthe non-negative orthantinto the positive orthant,however, then b hasjust one fixed point.

To see why, suppose b hasanother fixed point.

S

The Perron-Frobenius  Theorem In general, b can havemany fixed points.

If some power of B mapsthe non-negative orthantinto the positive orthant,however, then b hasjust one fixed point.

To see why, suppose b hasanother fixed point.

Consider the plane Mspanned by thecorresponding eigenrays,which intersects S in a linesegment m.

S

m

The Perron-Frobenius  Theorem The map B restricts to alinear operator on M.Correspondingly, b restrictsto a projectivetransformation of m.

If some power of B sendsthe non-negative orthantinto the positive orthant,b|m cannot be the identity.

Thus, the two fixed pointswe started with are theonly fixed points of b|m.They correspond to twodistinct eigenvalues of B|M.

m

The Perron-Frobenius  Theorem The fixed point of b|m withthe larger eigenvalue isattracting, and the one withthe smaller eigenvalue isrepelling.

m

The Perron-Frobenius  Theorem The fixed point of b|m withthe larger eigenvalue isattracting, and the one withthe smaller eigenvalue isrepelling.

When b|m acts, theendpoint of m next to therepelling fixed point will getrepelled clear out of thenon-negative orthant. Butthat's impossible!

Thus, imagining that bcould have more than onefixed point leads only toabsurdity.

m