# Contents. Outline. Structured Rank Matrices Lecture 2: The theorem Proofs Examples related to structured ranks References. Structure Transport

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1 Contents Structured Rnk Mtrices Lecture 2: Mrc Vn Brel nd Rf Vndebril Dept. of Computer Science, K.U.Leuven, Belgium Chemnitz, Germny, September Exmples relted to structured rnks 2 2 / 26 Outline 1 Exmples relted to structured rnks 2 Definition (Right null spce) Given mtrix A R m n. The right null spce N(A) equls Definition (Nullity of mtrix) N(A) = {x R n Ax = 0}. Given mtrix A R m n. The nullity n(a) is defined s the dimension of the right null spce of A. Corollry The dimension of the right null spce corresponds to the rnk deficiency of the columns of the mtrix A: n(a) = n rnk (A) = (number of columns) rnk (A). 3 / 26 4 / 26

2 Theorem (Nullity theorem) Suppose the following invertible mtrix A R n n is prtitioned s [ ] A11 A A = 12 A 21 A 22 with A 11 of size p q. The inverse B of A is prtitioned s [ ] A 1 B11 B = B = 12 B 21 B 22 with B 11 of size q p. Then the nullities n(a 11 ) nd n(b 22 ) re equl: n(a 11 ) = n(b 22 ). Corollry Corollries of the nullity theorem Suppose A R n n is nonsingulr mtrix, nd α, β re nonempty subsets of N with α < n nd β < n. Then rnk ( A 1 (α; β) ) = rnk (A(N\β; )) + α + β n. Proof: Permuting the mtrix such tht A(N\β; ) moves to the upper left position A 11, will move A 1 (α; β) to the position B 22. Using the equlities: gives us the proof. n(a 11 ) = n α rnk (A 11 ), n(b 22 ) = β rnk (B 22 ), 4 / 26 5 / 26 Corollry Corollries of the nullity theorem Suppose A R n n is nonsingulr mtrix, nd α, β re nonempty subsets of N with α < n nd β < n. Then rnk ( A 1 (α; β) ) = rnk (A(N\β; )) + α + β n. Corollry Corollries of the nullity theorem Suppose A R n n is nonsingulr mtrix, nd α, β re nonempty subsets of N with α < n nd β < n. Then rnk ( A 1 (α; β) ) = rnk (A(N\β; )) + α + β n. Exmples for 5 5 mtrices: α = {1, 2} nd N\β = {3, 4, 5} nd β = {1, 2} = {3, 4, 5} Exmples for 5 5 mtrices: α = {1, 2} nd N\β = {4, 5} nd β = {1, 2, 3} = {3, 4, 5} 5 / 26 5 / 26

3 Corollry Corollries of the nullity theorem Suppose A R n n is nonsingulr mtrix, nd α, β re nonempty subsets of N with α < n nd β < n. Then rnk ( A 1 (α; β) ) = rnk (A(N\β; )) + α + β n. Corollry Corollries of the nullity theorem Suppose A R n n is nonsingulr mtrix, nd α, β re nonempty subsets of N with α < n nd β < n. Then rnk ( A 1 (α; β) ) = rnk (A(N\β; )) + α + β n. Exmples for 5 5 mtrices: α = {3, 4, 5} nd N\β = {3, 4, 5} nd β = {1, 2} = {1, 2} Exmples for 5 5 mtrices: α = {2, 4} nd N\β = {2, 4, 5} nd β = {1, 3} = {1, 3, 5} 5 / 26 5 / 26 Some corollries of the nullity theorem Corollry For nonsingulr mtrix A R n n nd α N, we hve: rnk ( A 1 (α; ) ) = rnk (A(α; )). Some corollries of the nullity theorem Corollry For nonsingulr mtrix A R n n nd α N, we hve: rnk ( A 1 (α; ) ) = rnk (A(α; )). Proof: Is direct consequence of the previous eqution: ( ) rnk A 1 (α; β) = rnk (A(N\β; )) + α + β n, when posing β = : ( ) rnk A 1 (α; ) = rnk (A(α; )) + α + n. This mens tht for mtrix the following blocks lwys hve the sme rnk in A nd in A 1. α = {2, 3, 4, 5} nd α = {3, 4, 5} nd = {1} = {1, 2} 6 / 26 6 / 26

4 Some corollries of the nullity theorem Corollry For nonsingulr mtrix A R n n nd α N, we hve: rnk ( A 1 (α; ) ) = rnk (A(α; )). Some corollries of the nullity theorem Corollry For nonsingulr mtrix A R n n nd α N, we hve: rnk ( A 1 (α; ) ) = rnk (A(α; )). This mens tht for mtrix the following blocks lwys hve the sme rnk in A nd in A 1. α = {4, 5} nd α = {5} nd = {1, 2, 3} = {1, 2, 3, 4} This mens tht for mtrix the following blocks lwys hve the sme rnk in A nd in A 1. α = {3, 5} nd α = {2, 3} nd = {1, 2, 4} = {1, 4, 5} 6 / 26 6 / 26 Outline 1 Exmples relted to structured rnks 2 Different proofs There exist different strtegies to prove the nullity theorem. An importnt remrk, the theorem predicts structures but does not provide inversion formuls. Fiedler nd Mrkhm proved it, working directly on the rnks nd nullities of the blocks, their proof ws bsed on pper by Gustfson. Brrett nd Feinsilver were very close to n lterntive proof, but they only worked with tridigonl nd semiseprble mtrices. Recently lso Strng nd Nguyen proved weker formultion of the theorem. 7 / 26 8 / 26

5 Different proofs Proof (by Fiedler nd Mrkhm) Suppose n(a 11 ) n(b 22 ). If this is not true, we cn prove the theorem for the mtrices [ ] [ ] A22 A 21 B22 B, 21, A 12 A 11 B 12 B 11 which re lso ech others inverse. Suppose n(b 22 ) > 0 otherwise n(a 11 ) = 0 nd the theorem is proved. When n(b 22 ) = c > 0, then there exists mtrix F with c linerly independent columns, such tht B 22 F = 0. Remember tht [ ] [ ] [ ] A11 A 12 B11 B 12 I 0 =. A 21 A 22 B 21 B 22 0 I Different proofs Proof (by Fiedler nd Mrkhm) Hence, multiplying the following eqution to the right by F we get A 11 B 12 + A 12 B 22 = 0, Applying the sme opertion to the reltion: A 11 B 12 F = 0. (1) A 21 B 12 + A 22 B 22 = I it follows tht A 21 B 12 F = F, nd therefore rnk (B 12 F ) c. Using this lst sttement together with eqution (1), we derive n(a 11 ) rnk (B 12 F ) c = n(b 22 ). 9 / 26 With our ssumption n(a 11 ) n(b 22 ), this proves the theorem. 9 / 26 Outline 1 Exmples relted to structured rnks Exmple (Upper tringulr mtrix) The inverse of n upper tringulr mtrix is n upper tringulr mtrix / / 26

6 Exmple (Upper tringulr mtrix) The inverse of n upper tringulr mtrix is n upper tringulr mtrix. The rnk of the red mrked blocks is mintined by Corollry Exmple (Upper tringulr mtrix) The inverse of n upper tringulr mtrix is n upper tringulr mtrix. The rnk of the red mrked blocks is mintined by Corollry / / 26 Exmple (Upper tringulr mtrix) The inverse of n upper tringulr mtrix is n upper tringulr mtrix. The rnk of the red mrked blocks is mintined by Corollry Exmple (Upper tringulr mtrix) The inverse of n upper tringulr mtrix is n upper tringulr mtrix. The rnk of the red mrked blocks is mintined by Corollry / / 26

7 Exmple (Qusiseprble mtrix) The inverse of qusiseprble mtrix is qusiseprble mtrix. Exmple (Qusiseprble mtrix) The inverse of qusiseprble mtrix is qusiseprble mtrix. The rnk of the red mrked blocks is mintined by Corollry / / 26 Exmple (Qusiseprble mtrix) The inverse of qusiseprble mtrix is qusiseprble mtrix. The rnk of the red mrked blocks is mintined by Corollry 2. Exmple (Qusiseprble mtrix) The inverse of qusiseprble mtrix is qusiseprble mtrix. The rnk of the red mrked blocks is mintined by Corollry / / 26

8 Exmple (Qusiseprble mtrix) The inverse of qusiseprble mtrix is qusiseprble mtrix. The rnk of the red mrked blocks is mintined by Corollry 2. Exmple (Tridigonl vs. semiseprble) The inverse of tridigonl mtrix is semiseprble mtrix. 12 / / 26 Exmple (Tridigonl vs. semiseprble) The inverse of tridigonl mtrix is semiseprble mtrix. The rnk of the left block plus 1 equls the rnk of the right block, ccording to corollry 1 α = {3, 4, 5} nd N\β = {2, 3, 4, 5} nd β = {1} = {1, 2} Exmple (Tridigonl vs. semiseprble) The inverse of tridigonl mtrix is semiseprble mtrix. The rnk of the left block plus 1 equls the rnk of the right block, ccording to corollry 1 α = {4, 5} nd N\β = {3, 4, 5} nd β = {1, 2} = {1, 2, 3} / / 26

9 Exmple (Tridigonl vs. semiseprble) The inverse of tridigonl mtrix is semiseprble mtrix. The rnk of the left block plus 1 equls the rnk of the right block, ccording to corollry 1 α = {5} nd N\β = {4, 5} nd β = {1, 2, 3} = {1, 2, 3, 4} Exmple The inverse of {p, q}-semiseprble mtrix is {p, q}-bnd mtrix. One cn predict the structure of the inverse of generlized Hessenberg mtrix. One cn predict the structure when inverting hierrchiclly semiseprble nd/or H mtrices. Structure relted: The off-digonl structure is mintined. For exmple the inverse of rnk one mtrix plus digonl is gin rnk 1 mtrix plus digonl. Applicble to ll structured rnk mtrices. 13 / / 26 Outline for the nullity theorem 1 Exmples relted to structured rnks 2 W. H. Gustfson, A note on mtrix inversion, Liner Algebr nd Its Applictions 57 (1984), M. Fiedler, Bsic mtrices, Liner Algebr nd Its Applictions 373 (2003), W. W. Brrett, A theorem on inverse of tridigonl mtrices, Liner Algebr nd Its Applictions 27 (1979), W. W. Brrett nd P. J. Feinsilver, Gussin fmilies nd theorem on ptterned mtrices, Journl of Applied Probbility 15 (1978), W. W. Brrett nd P. J. Feinsilver, Inverses of bnded mtrices, Liner Algebr nd Its Applictions 41 (1981), G. Strng nd T. Nguyen, The interply of rnks of submtrices, SIAM Review 46 (2004), / / 26

10 Generl remrks LU nd QR-decompositions Given mtrix A R m n. A = LU is clled n LU-decomposition if L is lower tringulr nd U is upper tringulr. Frequently used for solving systems of equtions (Gussin elimintion). Computing eigenvlues of specilized mtrices (quotient-difference lgorithms). Generl remrks LU nd QR-decompositions Given mtrix A R m n. A = LU is clled n LU-decomposition if L is lower tringulr nd U is upper tringulr. Frequently used for solving systems of equtions (Gussin elimintion). Computing eigenvlues of specilized mtrices (quotient-difference lgorithms). Given mtrix A R m n. A = QR is clled QR-decomposition if Q is unitry (QQ H = Q H Q = I ) nd R is upper tringulr. Solving systems of equtions (more stble thn Gussin eliminition). In the top 10 lgorithms of the 20th century for computing eigenvlues of rbitrry mtrices. Under some mild conditions both fctoriztions re unique. Under some mild conditions both fctoriztions re unique. 17 / / 26 Outline 1 Exmples relted to structured rnks 2 Theorem (LU-fctoriztion) Given n invertible mtrix A, with LU fctoriztion A = LU. Let A be prtitioned s [ ] A11 A A = 12 A 21 A 22 with A 11 of dimension p q. Let U be prtitioned s [ ] U11 U U = 12 0 U 22 with U 11 of dimension p q. Then the nullities n(a 12 ) nd n(u 12 ) re equl (s well s their rnks). 18 / / 26

11 Outline Exmple (Structured rnk mtrices) The L nd U fctor inherit the structure. For semiseprble mtrix: U is upper semiseprble, nd L is lower semiseprble. For tridigonl mtrix: U is upper bidigonl, nd L is lower bidigonl. For {p, q}-semiseprble mtrix: U is {q}-upper semiseprble, nd L is {p}-lower semiseprble. For {p, q}-bnd mtrix: U is {q}-upper bnd, nd L is {p}-lower bnd. Holds for combintions, nd even more generl structures. 1 Exmples relted to structured rnks 2 20 / / 26 Theorem (QR-fctoriztion) Given n invertible mtrix A, with QR-fctoriztion A = QR. Let A be prtitioned s [ ] A11 A A = 12, A 21 A 22 with A 11 of dimension p q. Let Q be prtitioned s [ ] Q11 Q Q = 12, Q 21 Q 22 with Q 11 of dimension p q. Then the nullities n(a 21 ) nd n(q 21 ) re equl. Exmple (Structured rnk mtrices) The Q fctor inherits the structure of the lower tringulr prt. The structure of R is more complicted (see next slides). For semiseprble mtrix: Q hs the lower tringulr prt of lower semiseprble form, nd R hs the upper tringulr structure of rnk 2. For tridigonl mtrix: Q hs the lower tringulr prt of bidigonl form. For {p, q}-semiseprble mtrix: Q hs the lower tringulr prt of {p}-semiseprble form. For {p, q}-bnd mtrix: Q hs the lower tringulr prt of {p}-bnd form. Holds for combintions, nd even more generl structures. 22 / / 26

12 Rnk structure of the R-fctor We derive this structure by investigting how the originl rnk structure is trnsformed when computing the QR-fctoriztion. Rnk structure of the R-fctor We derive this structure by investigting how the originl rnk structure is trnsformed when computing the QR-fctoriztion. Strting sitution: Rk r Rk s Rnk structure of the R-fctor We derive this structure by investigting how the originl rnk structure is trnsformed when computing the QR-fctoriztion. First series of Givens trnsformtions: Rnk structure of the R-fctor We derive this structure by investigting how the originl rnk structure is trnsformed when computing the QR-fctoriztion. First series of Givens trnsformtions: Q 1 Rk r Rk s Rk s = Rk r Q 1 Rk s

13 Rnk structure of the R-fctor We derive this structure by investigting how the originl rnk structure is trnsformed when computing the QR-fctoriztion. First series of Givens trnsformtions: Rnk structure of the R-fctor We derive this structure by investigting how the originl rnk structure is trnsformed when computing the QR-fctoriztion. Second series of Givens trnsformtions: Q 2 Rk s = Rk r Q 1 Rnk structure of the R-fctor We derive this structure by investigting how the originl rnk structure is trnsformed when computing the QR-fctoriztion. Second series of Givens trnsformtions: Rnk structure of the R-fctor We derive this structure by investigting how the originl rnk structure is trnsformed when computing the QR-fctoriztion. Third series of Givens trnsformtions: Q 2 = Q 3

14 Rnk structure of the R-fctor We derive this structure by investigting how the originl rnk structure is trnsformed when computing the QR-fctoriztion. Third series of Givens trnsformtions: Rnk structure of the R-fctor We derive this structure by investigting how the originl rnk structure is trnsformed when computing the QR-fctoriztion. Third series of Givens trnsformtions: = Q 3 = Q 3 Q.E.D. Outline 1 Exmples relted to structured rnks for these generliztions R. Vndebril nd M. Vn Brel, A short note on the nullity theorem, Journl of Computtionl nd Applied Mthemtics 189: , / / 26

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