Matrix Theory -Matrix Inversion School of Mathematical Sciences Teaching Group Main Reference books Fuzhen Zhang.Matrix Theory-Basic Results and Techniques,Second Edition. Springer,2011. llse C.F.Ipsen,Numerical Matrix Analysis:Linear Systems and Least Squares. S1AM,2009. Reference books: o Roger A.Horn and Charles A.Johnson:Matrix Analysis.Cambridge University Press,1985. Gene H.Golub and Charles F.Van Loan:Matrix Computations,Third Edition. Johns Hopkins Press,1996. Nicholas J.Higham.Accuracy and Stability of Numerical Algorithms,Second Edition.SIAM,2002. Y.Saad.Iterative Methods for Sparse Linear Systems,Second Edition.SIAM, Philadelphia,2003. Matrix Theory Inverse Maintained by Yan-Fei Jing
: Main Reference books Fuzhen Zhang. Matrix Theory-Basic Results and Techniques, Second Edition. Springer, 2011. Ilse C. F. Ipsen, Numerical Matrix Analysis: Linear Systems and Least Squares. SIAM, 2009. Reference books: Roger A. Horn and Charles A. Johnson: Matrix Analysis. Cambridge University Press, 1985. Gene H. Golub and Charles F. Van Loan: Matrix Computations, Third Edition. Johns Hopkins Press, 1996. Nicholas J. Higham. Accuracy and Stability of Numerical Algorithms, Second Edition. SIAM, 2002. Y. Saad. Iterative Methods for Sparse Linear Systems, Second Edition. SIAM, Philadelphia, 2003. Maintained by Yan-Fei Jing Matrix Theory ––Matrix Inversion School of Mathematical Sciences Teaching Group Matrix Theory Inverse
Introduction Outline Introduction Existence of an Inverse ③Some Properties Sherman-Morrison Formula Inverse of a Partitioned Matrix Discovery Journey Comprehensive Problems 争老年这大习 Matrix Theory Inverse -2/35
Introduction Outline 1 Introduction 2 Existence of an Inverse 3 Some Properties 4 Sherman-Morrison Formula 5 Inverse of a Partitioned Matrix 6 Discovery Journey 7 Comprehensive Problems Matrix Theory Inverse - 2/35
Introduction Some observations If a is a nonzero scalar,then for each number B, the equation ax=B has a unique solution given by x=a1B. 奇电有这头 Matrix Theory Inverse -3/35
Introduction Some observations If α is a nonzero scalar, then for each number β, the equation αx = β has a unique solution given by x = α −1 β. To prove that α −1β is a solution, write α(α −1 β) = (αα −1 )β = (1)β = β. Uniqueness follows because if x1 and x2 are two solutions, then αx1 = β = αx2 ⇒ α −1 (αx1) = α −1 (αx2) ⇒ (α −1α)x1 = (α −1α)x2 ⇒ (1)x1 = (1)x2 ⇒ x1 = x2. Matrix Theory Inverse - 3/35
Introduction Some observations If a is a nonzero scalar,then for each number B, the equation aX=B has a unique solution given by x=a1B. o To prove that a-13 is a solution,write a(a-13)=(aa-1)3=(1)B=3. 奇电有这头 Matrix Theory Inverse -3/35
Introduction Some observations If α is a nonzero scalar, then for each number β, the equation αx = β has a unique solution given by x = α −1 β. To prove that α −1β is a solution, write α(α −1 β) = (αα−1 )β = (1)β = β. Uniqueness follows because if x1 and x2 are two solutions, then αx1 = β = αx2 ⇒ α −1 (αx1) = α −1 (αx2) ⇒ (α −1α)x1 = (α −1α)x2 ⇒ (1)x1 = (1)x2 ⇒ x1 = x2. Matrix Theory Inverse - 3/35
Introduction Some observations If a is a nonzero scalar,then for each number B, the equation aX=B has a unique solution given by x=a1B. To prove that a-13 is a solution,write a(a-13)=(aa-1)3=(1)B=3. o Uniqueness follows because if x1 and x2 are two solutions,then ax=B=a为→a-1(ax)=a-1(ax2) →(a1a)x=(a1a)x9 奇电有这女了 Matrix Theory (1)X1=(1)x9→xM=2: .3/35
Introduction Some observations If α is a nonzero scalar, then for each number β, the equation αx = β has a unique solution given by x = α −1 β. To prove that α −1β is a solution, write α(α −1 β) = (αα−1 )β = (1)β = β. Uniqueness follows because if x1 and x2 are two solutions, then αx1 = β = αx2 ⇒ α −1 (αx1) = α −1 (αx2) ⇒ (α −1α)x1 = (α −1α)x2 ⇒ (1)x1 = (1)x2 ⇒ x1 = x2. Matrix Theory Inverse - 3/35