# Eigenvalues of principle 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.)

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A counterexample (to the unedited question): $$A = \begin{pmatrix}1+x&1\\\1&1+x\end{pmatrix}.$$ Eigenvalues of $A$ are $2+x$ and $x$, principal minors have one eigenvalue $1+x$.

Voting to close.

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The poster clearly left out the condition that the matrix should be semidefinite and not definite (or else the interlacing inequalities make the condition impossible). –  Allen Knutson Mar 5 at 2:25
$$A= \begin{pmatrix} \frac{9}{2} & \frac{9}{20} & \frac{21}{20} & -\frac{3}{2}\\ -\frac{79}{11} & -\frac{3}{110} & \frac{23}{110} & \frac{31}{11}\\ \frac{6}{11} & \frac{21}{55} & \frac{114}{55} & \frac{6}{11}\\ \frac{16}{11} & \frac{12}{55} & \frac{128}{55} & \frac{5}{11} \end{pmatrix}$$ 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.