26-jordan

26-jordan - that we can have di²erent blocks with the same...

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JORDAN NORMAL FORM Math 21b, O. Knill JORDAN NORMAL FORM THEOREM. Every n × n matrix A is similar to a matrix [ A 1 ] 0 . . . 0 0 [ A 2 ] 0 0 . . . . . . . . . . . . 0 0 [ A k ] , where A i are matrices of the form A i = λ 1 0 0 0 0 λ 1 0 0 0 0 λ 1 0 0 0 0 λ 1 0 0 0 0 λ are matrices called Jordan blocks EXAMPLES OF MATRICES IN JORDAN FORM: 1) A generalized shear matrix A = ± 2 1 0 2 ² . It is itself a Jordan block. 2) 3 1 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 2 1 0 0 0 0 2 . This matrix has three Jordan blocks. The ±rst one is a 2x2 Jordan block, the second a 1x2 Jordan block, the third again a 2x2 Jordan block. 3) Every diagonal matrix like 3 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 2 0 0 0 0 0 2 is in Jordan normal form. It consists of 5 Jordan blocks. 4) 5 1 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 i 1 0 0 0 0 0 0 0 0 i 1 0 0 0 0 0 0 0 0 i 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 0 0 0 5 is in Jordan normal form. It consists of 4 Jordan blocks. Note that the diagonal elements can be complex and
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Unformatted text preview: that we can have di²erent blocks with the same diagonal elements. The eigenvalue 5 for example has here 3 Jordan blocks of size 1x1, 2x2 and 3x3. QUESTION: How many di²erent Jordan normal forms do exists for a 5 × 5 matrix with eigenvalues 3 and 2 of algebraic multiplicity 3 and 2? ANSWER: Examples 2) and 3) are examples. There are more. QUESTION: Is 3 3 3 1 2 2 . in Jordan normal form?...
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This note was uploaded on 04/06/2008 for the course MATH 21B taught by Professor Judson during the Spring '03 term at Harvard.

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