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 j1 ji (i) Prove diagonally dominant matrices are invertible. (ii) Prove that if A is diagonally dominant, then A  2max|aii | 1in

4. Prove that if A  Mm,p and B  Mp,n then rAB  rA  rB  n. n 5. Suppose A  Mn Define the function defined on Mn by A  max  |aij |. 1in j1 (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  MnR is column stochastic if all its elements are nonnegative and n  aji  1 for all i  1,,n j1 (i) Prove that the set of column stochastic matrices is multiplicatively closed.

(ii) Prove that the spectral radius A  1.

8. Let A,D  MnR 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  MnR 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 ckk1 and dkk1. 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  maxriA  i 1 th where as usual riA 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 x21 such that Hnx2  . 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.IftrA   0, k  1,,n, then A is nilpotent. 26. Let A  Mn.If  1  A we can define the Cayley transform of A by CA  I  A1I  A. Show that if A is skew-Hermitian, then CA is unitary and conversely. 27. Let A  Mn. Prove that if A is contained in the left half plane, then CA is n convergent. That is limmCA  0. (This is a little difficult.)

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