I am wondering if the following statement is true.
Let $A$ be an $n\times n$ Hermitian matrix and an eigenvalue $\lambda$ of $A$ have multiplicity $k$. Then, every $(n-k+1)\times (n-k+1)$ leading principal submatrix of $A$ has the eigenvalue $\lambda$.
That is correct. To make it more interesting, if leading principal submatrices are all positive, then the matrix is positive definite. The if statement can be changed to eigenvalues are all positive. This is what I learned in class. Yet I am unsure about the exact relationship between these two, I think probably it is fair to say leading principal minors are equivalent to eigenvalues.