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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 2017 • Question 1

Question

Math 6610 • Analysis of Numerical Methods I Fall 2017 • Question 1

Rank-One Perturbation of the Identity

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

Raw LaTeX

{Rank-One Perturbation of the Identity}

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

Topic Tags

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