Verify these matrix properties (easy and fun)

I’m looking at products like \(\mat{G}^t \mat{A} \mat{G}\) where the columns of \(\mat{G}\) are (nonisotropic) normal vectors. Specifically, I’d like to know the distribution of the eigen/singular-values of this product. Surprisingly, I was unable to find any results in the literature on this, so I started reading Gupta and Nagar to see if I could work it out using Jacobian magic.

In the preliminary material, they list some basic matrix algebra facts. Your mission, should you choose to accept it, is to prove the following:

  • If \(\mat{A} > \mat{0},\) \(\mat{B} > \mat{0},\) and \(\mat{A} – \mat{B} > \mat{0},\) then \(\mat{B}^{-1} – \mat{A}^{-1} > \mat{0}\)
  • If \(\mat{A}>\mat{0}\) and \(\mat{B} > \mat{0},\) then \(\det(\mat{A}+\mat{B}) > \det(\mat{A}) + \det(\mat{B})\)

The second’s easy, but for the first I found it necessary to use the fact that every positive matrix has a unique positive square root.

Some more, this time on Kronecker products:

  • If \(\mat{A}\) has eigenvalues \(\{\alpha_i\}\) and \(\mat{B}\) has eigenvalues \(\{\beta_j\},\) then \(\mat{A} \otimes \mat{B}\) has eigenvalues \(\{\alpha_i \beta_j\}.\)
  • If \(\mat{A}\) is \(m \times m\) and \(\mat{B}\) is \(n \times n\), then \(\det(\mat{A} \otimes \mat{B}) = \det(\mat{A})^n \det(\mat{B})^m\)

It’s useful here to note that
\[
(\mat{A} \otimes \mat{B}) (\mat{C} \otimes \mat{D}) = \mat{AC} \otimes \mat{BD},
\]
which has some obvious consequences, like that the Kronecker product of two orthogonal matrices is orthogonal. To be clear, the first Kronecker question can be addressed without any tedious calculations.

  • I should be studying for my Finals, but I could not resist peeking to see if you updated your blog site. It appears you have.

    With regards to the problems, what is a positive matrix?

  • swiftset

    One answer: if \(\langle x, \mat{A} x \rangle \geq 0\) for all nonzero vectors \(x\) then we say \(\mat{A}\) is positive (semidefinite). If the inequality is strict for all nonzero vectors, then we say \(\mat{A}\) is positive definite. It’s then easy to see that the set of positive matrices is a convex cone: \(\mat{A},\mat{B}\) positive implies any positive linear combination of the two is positive. This gives us a partial ordering on the set of matrices: \(\mat{A} \succeq \mat{B}\) iff \(\mat{A} – \mat{B}\) is positive.

    Sidenote: Basically everytime you hear someone talking about positive matrices, they really mean ‘symmetric positive semidefinite’ matrices. Technically there’s no requirement that the matrix be symmetric in the definition of positivity, that just happens to be the most used case.

  • swiftset

    Oh, good luck with your finals!