Forgive me, if I have made a blunder or missed something obvious, I'm not a mathematician!

I'm trying to understand 2 seemingly simple lines of maths - and understand how the conclusions are drawn from these. So here are the three lines (or rather two lines of maths, one statement at the end):

Let:

$L(n) := \{ A \in M(n) : A^T = A \}$

$P(n) := \{ A \in L(n) : A > 0 \} $

"The boundary of $P(n)$ is the set of singular positive semidefinite matrices"

So from what I understand here, the set L contains symmetric ($A^T = A$) n dimensional square matrices with real entries ($A \epsilon M(n)$). Then I have presumed that the part, $A > 0$, refers to the determinant of A, and not A itself.

I am also inferring (possibly incorrectly) that P(n) is therefore the space of SPD matrices..?

So why does ensuring we have square, real, symmetric matrices whose determinant is above 0 (this means they're invertible..?) ensure that the eigenvalues of said matrices are above 0 (my understanding of what makes a matrix positive definite)?

  • 1
    I think you are misreading $A>0$ as "$A$ has a positive determinant" when you should read it as "$A$ is strictly positive definite". The author might have written $A\succ 0$ to make this somewhat clearer. – kimchi lover Dec 6 at 15:30
  • Unfortunately the author used >, but you may be right! If so then my question is rather moot. Would it not make any sense to draw the SPD conclusions from my original interpretation? – bidby Dec 6 at 15:34
  • It's all nonsense under the determinant interpretation (the 2 by 2 diagonal matrix with -1, -1 on the diagonal has positive determinant and is not PSD or PD; the diagonal matrix with -1 and 0 on the diagonal is not in the closure of the PSD matrices). Under the other interpretation it all makes sense, in a trite way. – kimchi lover Dec 6 at 15:42

As others have stated, $P(n)$ is intended to represent the set of all positive definite $n \times n$ matrices (not the set of all matrices with positive determinant, as bidby initially thought). We let $A \succ 0$ denote a positive definite matrix and $A \succeq 0$ denote a positive semidefinite matrix.

Claim: $\partial P(n) \overset{\textrm{Def}}{=} \overline{P(n)} \setminus \mathring{P(n)} = \{A \in \mathbb{R}^{n \times n} | A \succeq 0, \nexists A^{-1}\}$.

Proof: Let $A \in \partial P(n)$, $(A_k)_{k \in \mathbb{N}} \subseteq P(n)$, and $z \in \mathbb{R}^n$. Then $$ z^T A z = z^T \left( \lim_{k \to \infty} A_k \right) z = \lim_{k\to\infty} \underbrace{\left( z^T A_k z \right)}_{\geq 0} \geq 0, $$ since the functions $f : \mathbb{R}^n \to \mathbb{R}, \; x \mapsto z^T x$ and $g : \mathbb{R}^{n \times n} \to \mathbb{R}^n, \; B \mapsto B z^T$ are linear (and linear transformations between finite dimensional vector spaces are continuous). Therefore, $A$ is positive semidefinite.

We now show that $A$ is singular. Since $A$ is positive semidefinite, all its eigenvalues are non-negative. We know that $A$ is not positive definite, since $\mathring{P(n)} = P(n)$ by this post. But if the eigenvalues were all positive, then $A$ would be positive definite. Hence, $A$ has $0$ as an eigenvalue -- i.e. $\exists$ eigenvector $v \in \mathbb{R}^n \setminus \{0\}$ with $Av = 0 \cdot v = 0$. Therefore, $\textrm{Ker}(A) \neq \{0\}$, so $A$ is singular (see 3. here).

(A trivial example of such an $A$ and such a sequence $(A_n)$ is $A := 0 \in \mathbb{R}^{1 \times 1}$ and $A_n := 1/n$.)

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