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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Spring 2022 • Question 1

Question

Math 6610 • Analysis of Numerical Methods I Spring 2022 • 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