Numerical Linear Algebra with Applications

15.10 Problems

15.1 Prove that for all rank-one matrices, ${\sigma }_1^2={\displaystyle {\sum }_{i=1}^m{\displaystyle {\sum }_{j=1}^n{a}_{ij}^2}}$. Hint: Use Theorem 15.5.

15.2 Suppose u₁, u₂, …, u_n and v₁, v₂, …, v_n are orthonormal bases for Ρⁿ. Construct the matrix A that transforms each v_i into u_i to give Av₁ = u₁, Av₂ = u₂, …, Av_n = u_n.

15.3 Let $A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ 0& a_{22}\end{array}\right]$. Determine value(s) for the a_ij so that A has distinct singular values.

15.4 Prove that $\text{rank}\left(A^{\text{T}}A\right)=\text{rank}\left(A{A}^{\text{T}}\right)$.

15.5 Find the SVD of

a. A^TA.

b. ${\left(A^{\text{T}}A\right)}^{-1}$