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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: that we can have dierent 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 dierent 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?...
View Full Document

Ask a homework question - tutors are online