Deducing that a matrix is indefinite using only its leading principal minors

Question

$A$ is indefinite iff $A$ fits none of the above criteria. Equivalently, $A$ has both positive and negative eigenvalues. Also equivalently, $x^TAx$ is positive for at least one $x$ and negative for at least another $x$.

Note that the leading principal minors refer to the determinants of the northwest-corner submatrices, and are merely a subset of all the principal minors.

Now, suppose that a symmetric $n\times n$ matrix $M$ is neither positive definite nor negative definite. Then, can we deduce the following statement (2) or at least statement (1)?

1. If $M$'s leading principal minors are all nonzero, then $M$ is indefinite;
2. If $M$ has some nonzero leading principal minor, then $M$ is indefinite.

I have seen both assertions separately in different texts (e.g. http://people.ds.cam.ac.uk/iar1/teaching/Hessians-DefinitenessTutorial.pdf and http://www.econ.ucsb.edu/~tedb/Courses/GraduateTheoryUCSB/BlumeSimonCh16.PDF), but am unable to prove either.

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EDIT

• If a symmetric $n\times n$ matrix $M$ is neither positive definite nor negative definite and det $M$ is nonzero, then $M$ is indefinite.

Proof: If det M is nonzero then we can immediately deduce that $M$ has no zero eigenvalues, and since $M$ is neither positive definite nor negative definite, $M$ can only be indefinite.

Answer 1

Edited per Ryan's clarification below.

Statement 1: Yes, this is fine. If $M$ is neither positive nor negative definite, and has no zero eigenvalues, then it must have at least one positive and one negative eigenvalue. Notice that this is a sufficient but not necessary condition on $M$ being indefinite. $\left[\begin{array}{ccc}0 & 0 &0\\0 & 1 & 0\\0 & 0 & -1\end{array}\right]$ is indefinite, for instance.

Statement 2: No, this is false. Consider for instance $\left[\begin{array}{cc}1 & 0\\0 & 0\end{array}\right]$ which is positive-semidefinite.

It is impossible to characterize indefinite matrices from the leading minors alone. For example, if the first row and column of a symmetric matrix $M$ is zero, the matrix might be positive-semidefinite, negative-semidefinite, or indefinite, yet all of the leading minors will be zero.

A complete, correct statement requires looking at all principal minors, for example: a symmetric matrix $M$ is indefinite (has positive and negative eigenvalues) if and only if:

1. $\Delta_k < 0$ for some even $k$; or
2. $\Delta_{k_1} > 0$ and $\Delta_{k_2} < 0$ for two different odd $k_1$ and $k_2$.

Knowing that $M$ is not strictly positive- or negative-definite does not really help. You can check that if $M$ satisfies neither of these conditions, then it must satisfy one of the rows of the purple box.

EDIT: Proof of the "only if" direction. Let $M$ be indefinite. Suppose, for contradiction, that neither of the above two hold. Then either all of the odd-dimensional minors are nonnegative, or all are nonpositive.

In the former case, $M$ satisfies the third row of the purple box above, and $M$ is positive-semidefinite, a contradiction.

In the latter case, $M$ satisfies the fourth row of the purple box above, and $M$ is negative-semidefinite, a contradiction.

EDIT 3: Proof of the "if" direction. Suppose one of the even-dimensional minors is negative, and suppose, for contradiction, that $M$ is positive-semidefinite or negative-semidefinite. Then by row three or four of the purple box (as appropriate), that minor is in fact positive, a contradiction. Therefore $M$ is neither positive- nor negative-semidefinite, and so is indefinite.

Suppose instead one of the odd-dimensional minors is positive, and another is negative, and suppose $M$ is positive-semidefinite. Then both of those minors are positive, a contradiction. Now suppose $M$ is negative-semidefinite. Then both of those minors are negative, a contradiction. The only remaining possibility is that $M$ is indefinite.