Math 640 — Midterm practice examination 1. Give a proof that if A, B Mn and AB has rank n, then A has rank n. 2 2. We say a matrix A Mn has a square root B Mn if B A. Prove that every diagonalizable matrix has a square root. 3. We say a matrix A Mn is diagonally dominant if n |aij | |aii | for all i 1,,n j1 ji (i) Prove diagonally dominant matrices are invertible. (ii) Prove that if A is diagonally dominant, then A 2max|aii | 1in 4. Prove that if A Mm,p and B Mp,n then rAB rA rB n. n 5. Suppose A Mn Define the function defined on Mn by A max |aij |. 1in j1 (i) Prove that is a norm on Mn. n (ii) Prove that is the matrix norm pertaining the norm on . 1 6. Suppose A Mn and there is an invertible matrix S Mn, for which S AS D is a diagonal matrix. Prove that the columns of S are the eigenvectors of A. (Give a complete proof.) 7. We say a matrix A MnR is column stochastic if all its elements are nonnegative and n aji 1 for all i 1,,n j1 (i) Prove that the set of column stochastic matrices is multiplicatively closed. (ii) Prove that the spectral radius A 1. 8. Let A,D MnR where D is diagonal with distinct diagonal elements. Prove that A and D commute if and only if A is also diagonal. 9. Let A MnR be skew-symmetric. Show that A is diagonalizable. 10. Prove that Unitary matrices are closed in the pointwise topology. That is, if Un is a sequence of unitary matrices and limUn V for each entry, then V is unitary. 1 11. Which Householder transformations on 2 are rotations? 12. Show that for every matrix in A Mn there is a lower triangular matrix L and an upper triangular matrix U such that LA U 13. Prove that the unitary matrices are closed the norm. That is, if the sequence {U_n} Mn are all unitary and limn Un U in the 2 norm, then U is also unitary. n n 14. Given two finite sequences ckk1 and dkk1. Prove that k|ckdk | maxk|dk|k|ck |. 15. Verify that a matrix norm which is subordinate to a vector norm satisfies norm conditions (i) and (ii). 16. Let A Mn. Show that the matrix norm subordinate to the vector norm is given by A maxriA i 1 th where as usual riA denotes the i row of the matrix A and 1 is the .1 norm. 17. The Hilbert matrix, Hn of order n is defined by 1 hij 1 i, j n i j 1 18. Prove or disprove that Hn 1 lnn. 19. Show that Hn 1. 1 2 20. Show that Hn 2 n as n . 21. Show that the spectral radius of Hn is bounded by 1. 22. Show that for each 0 there exists an integer N such that if n N there is a vector x Rn with x21 such that Hnx2 . 23. Same as the previous question except that you need to show that N 1 1 will 1/2 work. 24. Prove that the product of two isometries is an isometry. k n 25. Let A Mn.IftrA 0, k 1,,n, then A is nilpotent. 26. Let A Mn.If 1 A we can define the Cayley transform of A by CA I A1I A. Show that if A is skew-Hermitian, then CA is unitary and conversely. 27. Let A Mn. Prove that if A is contained in the left half plane, then CA is n convergent. That is limmCA 0. (This is a little difficult.) 2.