# Edu and ji a n d e ai iki ji of course an even

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Unformatted text preview: . . , λs }, (7.8.3) D E and where J(λ2 ) = diag (J1 (λ2 ), J2 (λ2 ), . . . , Jt2 (λ2 )) is a Jordan segment composed of Jordan blocks J (λ2 ) with the following characteristics. ⎞ ⎛λ 1 2 ⎜ Each Jordan block in J(λ2 ) has the form J (λ2 ) = ⎜ ⎝ .. . .. . .. . T H 1 λ2 ⎟ ⎟. ⎠ There are t2 = dim N (A − λ2 I) Jordan blocks in segment J(λ2 ). The number of i × i Jordan blocks in segment J(λ2 ) is νi (λ2 ) = ri−1 (λ2 ) − 2ri (λ2 ) + ri+1 (λ2 ), where ri (λ2 ) = rank (A − λ2 I)i . IG R I 0 If we set P2 = Q1 0 Q , then P2 is a nonsingular matrix such that 2 ⎞ ⎛ 0 0 J(λ1 ) P−1 AP2 = ⎝ 0 J(λ2 ) 0 ⎠ , where σ (A2 ) = {λ3 , λ4 , . . . , λs }. 2 0 0 A2 Y P Repeating this process until all eigenvalues have been depleted results in a nonsingular matrix Ps such that P−1 APs = J = diag (J(λ1 ), J(λ2 ), . . . , J(λs )) s in which each J(λj ) is a Jordan segment containing tj = dim N (A − λj I) Jor79 dan blocks. The matrix J is called the Jordan form for A (some texts refer to J as the Jordan canonical form or the Jordan normal form). The Jordan structure of A is deﬁned to be the number of Jordan segments in J along with the number and sizes of the Jordan blocks within each segment. The proof of uniqueness of the Jordan form for a nilpotent matrix (p. 580) can be extended to all A ∈ C n×n . In other words, the Jordan structure of a matrix is uniquely determined by its entries. Below is a formal summary of these developments. O C 79 Marie Ennemond Camille Jordan (1838–1922) discussed this idea (not over the complex numbers but over a ﬁnite ﬁeld) in 1870 in Trait´ des substitutions et des ´quations algebraique e e that earned him the Poncelet Prize of the Acad´mie des Science. But Jordan may not have e been the ﬁrst to develop these concepts. It has been reported that the German mathematician Karl Theodor Wilhelm Weierstrass (1815–1897) had previously formulated results along these lines. However, Weierstrass did not publish his ideas because he was fanatical about rigor, and he would not release his work until he...
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