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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Spring 2020 • Question 1

Question

Math 6610 • Analysis of Numerical Methods I Spring 2020 • Question 1

Rank-One Perturbation of the Identity

If \(x\) and \(y\) are \(m\)-vectors, the matrix \(A=I+xy^*\) is known as a rank-one perturbation of the identity. Show that if \(A\) is nonsingular, then its inverse has the form \(A^{-1}=I+\beta xy^*\) for some scalar \(\beta \), and give an expression for \(\beta \). For what \(x\) and \(y\) is \(A\) singular? If it is singular, what is \(\operatorname {null}(A)\)?

Raw LaTeX

{Rank-One Perturbation of the Identity}

If \(x\) and \(y\) are \(m\)-vectors, the matrix \(A=I+xy^*\) is known as a rank-one
perturbation of the identity. Show that if \(A\) is nonsingular, then its inverse
has the form \(A^{-1}=I+\beta xy^*\) for some scalar \(\beta\), and give an
expression for \(\beta\). For what \(x\) and \(y\) is \(A\) singular? If it is singular,
what is \(\operatorname{null}(A)\)?

Topic Tags

Conditioning, Sensitivity, and Perturbation Bounds Linear Programming and Optimization Matrix Rank and Fundamental Linear Algebra Singular Value Decomposition