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I need to find an expression for determining the eigenvalues of this matrix block $$ Acc=\begin{bmatrix} A & I \\ L & 0 \\ \end{bmatrix} $$ where

  • $A$ is a matrix of size $n \times n$
  • $L$ is a symmetric matrix of size $n \times n$
  • $0$ is the null matrix of size $n \times n$
  • $I$ is the identity matrix of size $n \times n$

Because the size of these matrices is very large and I have to carry out a program for calculating these eigenvalues I wish I could find a relationship relating the eigenvalues of $Acc$ and the eigenvalues of $A$ and $L$.

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  • $\begingroup$ There is some material here about Determinants of Block Matrices, and after reading it would seem to me that knowing the invertibility of $A$ is helpful..., though I'm not sure. $\endgroup$ – Eleven-Eleven Dec 8 '15 at 18:31
  • $\begingroup$ Thanks, I had noticed that the question was not clear and I have update it for more details: $\endgroup$ – Luca Angelino Dec 8 '15 at 21:29
  • $\begingroup$ Because the size of this matrix are very large and have to carry out a program for calculating these eigenvalues I wish I could find a relationship Brother eigenvalues of Acc and the eigenvalues of A and L $\endgroup$ – Luca Angelino Dec 8 '15 at 21:30
  • $\begingroup$ with the properties of the determinants of wikipedia I have not been able to get useful results $\endgroup$ – Luca Angelino Dec 8 '15 at 21:31

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