<s>[INST] Eigenvalues of principal minors Vs. eigenvalues of the matrix Say I have a positive semi-definite matrix with least positive eigenvalue x. Are there always principal minors of this matrix with eigenvalue less than x?
(Here "semidefinite" can not be taken to include the case "definite" -- there should be a
zero eigenvalue.)
 [/INST] No, it is not true;
Consider the matrix
$$A= 
\begin{pmatrix} 
\frac{9}{2} & \frac{9}{20} & \frac{21}{20} & -\frac{3}{2}\\
-\frac{79}{11} & -\frac{3}{110} & \frac{23}{110} & \frac{31}{11}\\
\frac{6}{11} & \frac{21}{55} & \frac{114}{55} & \frac{6}{11}\\
\frac{16}{11} & \frac{12}{55} & \frac{128}{55} & \frac{5}{11}
\end{pmatrix}
$$
which has eigenvalues 0,1,2,3 (so it is positive semi-definite, but not definite.)
The four principal minors are
$$\frac{27}{22},\frac{189}{110},\frac{153}{55},\frac{36}{11}$$
sorted in increasing order. This should give a definite negative answer to your question.
</s>