Kroneckers small physical size was another sensitive

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Unformatted text preview: 2 −1 0 −1 0 0 0 2 0 0 ⎞ 4 3 5 1⎟ ⎟ 2 0⎟ ⎟. −12 −7 ⎟ ⎠ −1 0 −9 −5 Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.8 Jordan Form http://www.amazon.com/exec/obidos/ASIN/0898714540 591 Solution: Computing the eigenvalues (which is the hardest part) reveals two distinct eigenvalues λ1 = 2 and λ2 = −1, so there are two Jordan segments in 0 the Jordan form J = J(2) J(−1) . Computing ranks ri (2) = rank (A − 2I)i 0 and ri (−1) = rank (A + I)i until rk ( ) = rk+1 ( ) yields r1 (2) = rank (A − 2I) = 4, r1 (−1) = rank (A + I) r2 (2) = rank (A − 2I) = 3, r2 (−1) = rank (A + I) r3 (2) = rank (A − 2I) = 2, r4 (2) = rank (A − 2I) = 2, It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] 2 3 4 = 4, 2 = 4, D E so k1 = index (λ1 ) = 3 and k2 = index (λ2 ) = 1. This tells us that the largest Jordan block in J(2) is 3 × 3, while the largest Jordan block in J(−1) is 1 × 1 so that J(−1) is a diagonal matrix (the associated eigenvalue is semisimple whenever this happens). Furthermore, T H ν3 (2) = r2 (2) − 2r3 (2) + r4 (2) = 1 =⇒ one 3 × 3 block in J(2), ν2 (2) = r1 (2) − 2r2 (2) + r3 (2) = 0 =⇒ no 2 × 2 blocks in J(2), ν1 (2) = r0 (2) − 2r1 (2) + r2 (2) = 1 =⇒ one 1 × 1 block in J(2), IG R ν1 (−1) = r0 (−1) − 2r1 (−1) + r2 (−1) = 2 =⇒ two 1 × 1 blocks in J(−1). ⎛ 2 1 0 ⎜0 2 1 Therefore, J(2) = ⎝ 0 0 2 ⎞ 0 0⎟ 0⎠ Y P 0 0 0 2 ⎛ O C J= J(2) 0 0 J(−1) ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎝ and J(−1) = −1 0 0 −1 210 021 002 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 −1 0 0 0 0 0 so that 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ 0 ⎟. ⎟ ⎟ ⎟ 0⎠ −1 The above example suggests that determining the Jordan form for An×n is straightforward, and perhaps even easy. In theory, it is—just find σ (A) , and calculate some ranks. But, in practice, both of these tasks can be difficult. To begin with, the rank of a matrix is a discontinuous function of its entries, and r...
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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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