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Is there any relation between these theorems on interlacing roots?

  • The roots of $f(x), f'(x)$ interlace (if all the roots of $f(x)$ are real and have real coefficients).
  • The eigenvalues of an $n \times n$ matrix $A$ and it's minor $n-1\times n-1$ matrix $A_{ii}$ interlace (is a real symmetric matrix, or Hermitian).

Is there a more modern version of these results?

EDIT: Picturing Terry Tao's answer, let $f(x)= (x-1)(x-2)\dots(x-8)(x-9)$.

A plot of $\{(f(x), f'(x)): 0.5 < x < 9\}$. This figure is symmetric, and not very representative.

alt text
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Is there any relation between these theorems on interlacing roots?

  • The roots of $f(x), f'(x)$ interlace (if all the roots of $f(x)$ are real and have real coefficients).
  • The eigenvalues of an $n \times n$ matrix $A$ and it's minor $n-1\times n-1$ matrix $A_{ii}$ interlace (is a real symmetric matrix, or Hermitian).

Is there a more modern version of these results?

EDIT: Picturing Terry Tao's answer, let $f(x)= (x-1)(x-2)\dots(x-8)(x-9)$.

A plot of $\{(f(x), f'(x)): 0.5 < x < 9\}$. This figure is symmetric, and not very representative.

alt text
show/hide this revision's text 5 Only principle minors

Is there any relation between these theorems on interlacing roots?

  • The roots of $f(x), f'(x)$ interlace (if all the roots of $f(x)$ are real and have real coefficients).
  • The eigenvalues of an $n \times n$ matrix $A$ and it's minor $n-1\times n-1$ matrix $A_{ij}$ A_{ii}$ interlace (is a real symmetric matrix, or Hermitian).

Is there a more modern version of these results?
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