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College Algebra Exam Review 401

# College Algebra Exam Review 401 - 8.7 JORDAN CANONICAL FORM...

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8.7. JORDAN CANONICAL FORM 411 Let A be an n –by– n matrix over K , and suppose that the characteristic polynomial of A factors into linear factors in KOExŁ . Let T be the linear transformation of K n determined by left multiplication by A . Thus, A is the matrix of T with respect to the standard basis of K n . A second matrix A 0 is similar to A if, and only if, A 0 is the matrix of T with respect to some other ordered basis of K n . By Theorem 8.7.4 A is similar to a matrix A 0 in Jordan canonical form, and the matrix A 0 is unique up to permutation of Jordan blocks. W have proved: Theorem 8.7.6. (Jordan canonical form for matrices.) Let A be an n by– n matrix over K , and suppose that the characteristic polynomial of A factors into linear factors in KOExŁ . Then A is similar to a matrix in Jordan canonical form, and this matrix is unique up to permuation of the Jordan blocks. Definition 8.7.7. A matrix is called the Jordan canonical form of A if it is in Jordan canonical form, and it is similar A .
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