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/Solutions

1. Diagonal matrices

A matrix A is a diagonal matrix if it is a square matrix with Aij=0 whenever i≠j.

  1. Prove or disprove: If A and B are diagonal matrices of the same size, so is AB.
  2. Let p(A)=Πi Aii. Prove or disprove: If A and B are diagonal matrices as above, then p(AB) = p(A)p(B).

2. Matrix square roots

  1. Show that there exists a matrix A such that A≠0 but A²=0.
  2. Show that if A²=0, there exists a matrix B such that B²=I+A. Hint: What is (I+A)²?

3. Dimension reduction

Let A be an n×m random matrix obtained by setting each entry Aij independently to ±1 with equal probability.

Let x be an arbitrary vector of dimension m.

Compute E[||Ax||²], as a function of ||x||, n, and m, where ||x|| = (x⋅x)1/2 is the usual Euclidean length.

4. Non-invertible matrices

Let A be a square matrix.

  1. Prove that if Ax=0 for some column vector x≠0, then A-1 does not exist.

  2. Prove that if the columns of A are not linearly independent, then A-1 does not exist.

  3. Prove that if the rows of A are not linearly independent, then A-1 does not exit.


2014-06-17 11:57