By David M. Young
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Additional info for Iterative Solution of Large Linear Systems
Let A' — ε~χΑ and let V be any matrix which reduces A' to Jordan canonical form. 43) where J(e) is the same as the Jordan canonical form J of A except that the off-diagonal elements of J are multiplied by ε. Thus a typical block of/(e> is 'λ ε 0 . 44) λ ε 0 λ, Since EE1 has the form 1 0 0 0 1 0 0 ON 0 0 ΕΕτ = ■(145) 0 0 0 ,0 0 0 1 0 0 0> it follows that 5(££:T) = 1 and \\E\\ = 1. 46) This theorem is given by Householder . The proof is based on that of House holder but with the use of a technique of Ortega and Rheinboldt .
Since 5 is similar to ß , μ is an eigenvalue of 5 and Â is not positive definite. 8, Â must be positive definite. This contradiction proves that μ < 1 and shows that A is an M-matrix. The class of M-matrices is a subclass of the class of monotone matrices. 3. A matrix A is a monotone matrix, if A is nonsingular and A-1 > 0. 4. A matrix A is monotone if and only if Ax > 0 implies *>0. Proof. Suppose A is monotone. If Ax = y > 0, then x = ^4 -1 y > 0. On the other hand, suppose Ax > 0 implies x > 0, and let # be any vector such that Az = 0.
29) (We note that w φ 0; otherwise, 0^· = 0 for ally and A = 0). 5) follows. 6) is the induced matrix norm corresponding to the vector norm || · l^. 2 that the last expression is equal to (S(A*A))i. 7). This latter norm is sometimes referred to as the spectral norm. When no confusion will arise we shall omit the norm labels. 32) A vector norm || · \\x and a matrix norm || · ||^ are said to be consistent, or compatible, if for all v e EN we have || Av ||. 33) Evidently, any vector norm and the induced matrix norm are consistent.