8.7. JORDAN CANONICAL FORM411LetAbe ann–by–nmatrix overK, and suppose that the characteristicpolynomial ofAfactors into linear factors inKOExŁ. LetTbe the lineartransformation ofKndetermined by left multiplication byA. Thus,Aisthe matrix ofTwith respect to the standard basis ofKn. A second matrixA0is similar toAif, and only if,A0is the matrix ofTwith respect to someother ordered basis ofKn. By Theorem8.7.4Ais similar to a matrixA0in Jordan canonical form, and the matrixA0is unique up to permutation ofJordan blocks. W have proved:Theorem 8.7.6.(Jordan canonical form for matrices.) LetAbe ann–by–nmatrix overK, and suppose that the characteristic polynomial ofAfactors into linear factors inKOExŁ. ThenAis similar to a matrix in Jordancanonical form, and this matrix is unique up to permuation of the Jordanblocks.Definition 8.7.7.A matrix is called theJordan canonical form ofAifit is in Jordan canonical form, andit is similarA.
This is the end of the preview.
access the rest of the document.