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Transcript of wyklad3 - IBS PANszulck/PHD/wyklad3.pdfHUMAN CAPITAL all apli dy(t) EUROPEAN UNION EUROPEAN SOCIAL...
Jordan canonical form:distinct eigenvalues
๐๐(๐ก)๐๐ก = ๐๐ ๐ก , transformation of ๐(๐ก) to
๐ ๐ก = ๐โ1๐(๐ก) ๐ - a nonsingular matrix
๐๐(๐ก)๐๐ก = ๐ฒ๐ ๐ก ๐ฒ = ๐โ1๐๐๐ฒ - Jordan canonical form
The form depends on eigenvalues of ๐๐ โ ๐๐ ๐ = ๐det ๐ โ ๐๐ = 0 โ ๐ eigenvalues ๐๐๐๐ - eigenvectors
distinct (single) eigenvalues:
columns of ๐ are eigenvectors, as๐๐ = ๐๐ฒ๐ ๐1,โฆ , ๐๐ = [๐1๐1,โฆ , ๐๐๐๐]๐ โ ๐1๐ ๐1 = ๐,โฆ , ๐ โ ๐๐๐ ๐๐ = ๐
For distinct eigenvalues ๐๐ โ ๐๐ for ๐ โ ๐
๐ฒ =๐1 0
โฑ0 ๐๐
๐๐ฒ๐ก =๐๐1๐ก 0
โฑ0 ๐๐๐๐ก
๐ ๐ก = ๐๐ ๐ก =๐ 11๐๐1๐ก โฏ ๐ 1๐๐๐๐๐กโฎ โฑ โฎ
๐ ๐1๐๐1๐ก โฏ ๐ ๐๐๐๐๐๐ก๐ ๐ก - a fundamental matrix (Example (ยฃ))
a general solution
of the homogeneous equation
for distinct eigenvalues
๐0 ๐ก = ๐ ๐ก ๐ = ๐=1๐ โ๐๐๐๐๐๐๐ก = ๐=1๐ ๐๐๐๐๐๐ก
the characteristic equation
๐๐ =๐ ๐1โฎ๐ ๐๐
โ ๐
๐ ๐ก = ๐๐ฒ๐ก
Institute of Computer Science PAS
Warsaw, 21.03.2013
The Project is co-financed by the European Unionfrom resources of the European Social Found