Calculation of the Weyr Characteristic from the Singular Graph of an M-matrix |
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Page 34
... following are equivalent : ( i ) for all A ɛ a ( § ) , w 。( A ) = 1p ' ( ii ) for all p , p = = P 1 , ... , h ; 1 , ... , h - 1 , there do not exist non - empty disjoint subsets $ 1 , $ 2 በ ΠΛ ^ ( 1 ) ^ p = ^ ( 82 ) ^^ · of A p + 1 ...
... following are equivalent : ( i ) for all A ɛ a ( § ) , w 。( A ) = 1p ' ( ii ) for all p , p = = P 1 , ... , h ; 1 , ... , h - 1 , there do not exist non - empty disjoint subsets $ 1 , $ 2 በ ΠΛ ^ ( 1 ) ^ p = ^ ( 82 ) ^^ · of A p + 1 ...
Page 40
... following are equivalent for p , 1 ≤ p ≤ h : ( i ) w1 + ... + W р = + + λ р ; Q.E.D. ( ii ) Ker AP has a basis of semi - positive vectors . Proof : ( i ) ( ii ) . By ( 4.3 ) and ( 4.4 ) , ( i ) implies that E = > Ep = Ker AP . By ...
... following are equivalent for p , 1 ≤ p ≤ h : ( i ) w1 + ... + W р = + + λ р ; Q.E.D. ( ii ) Ker AP has a basis of semi - positive vectors . Proof : ( i ) ( ii ) . By ( 4.3 ) and ( 4.4 ) , ( i ) implies that E = > Ep = Ker AP . By ...
Page 41
... following result completes the Main Theorem ( 4.7 ) . ( 6.5 ) Theorem : Let A be an M - matrix with singular graph S and level numbers ( ^ 1 , ... , ) . The following are equivalent : i ) the Weyr characteristic for A ( associated with ...
... following result completes the Main Theorem ( 4.7 ) . ( 6.5 ) Theorem : Let A be an M - matrix with singular graph S and level numbers ( ^ 1 , ... , ) . The following are equivalent : i ) the Weyr characteristic for A ( associated with ...
Contents
2 Standard Forms and the Singular Graph S | 4 |
3 Construction of an Sbasis for EA | 11 |
4 Relationship between wA and an Smatrix for A | 22 |
4 other sections not shown
Common terms and phrases
a-th column associated scalars B₁ B₂ block column ccrA combinatorially independent Corollary Daniel James Richman define Definition denote diagonal blocks DP+1 eigenspace E(A elementary divisor exist non-empty disjoint finite partially ordered following are equivalent full column rank Hence inductive assumption Jordan basis Ker AP Lemma Lemna Let vª level numbers linearly dependent linearly ordered M-matrix with singular Main Theorem non-empty disjoint subsets non-negative non-singular Observe p-th level partially ordered set permutation matrix Proof prove Q. E. D. Remark result follows Rothblum S-basis for E(A S-set satisfies condition 5.6ii Schneider 1956 semi-positive set with level singular graph singular matrix ß ɛ standard form suppose Theorem 4.7 thesis University of Wisconsin-Madison v₁ vectors w₁ Weyr characteristic αε αελ αεφ βα βε βελ γα λ₁ ΠΛ га