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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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