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Robert Israel
Robert Israel
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Let's write this in block matrices: $$B = \pmatrix{B_{11} & 0\cr 0 & 0\cr}, \ A = \pmatrix{A_{11} & A_{12}\cr A_{21} & A_{22}\cr}, u = \pmatrix{u_1 \cr u_2\cr}$$ where $B_{11}$ has full rank. Then the eigenvector equations $A u = \lambda B u$ become $A_{11} u_1 + A_{12} u_2 = \lambda B_{11} u_1$ and $A_{21} u_1 + A_{22} u_2 = 0$. Suppose $A_{22}$ is invertible. Then we have $u_2 = - A_{22}^{-1} A_{21} u_1$, and $(A_{11} - A_{12} A_{22}^{-1} A_{21}) u_1 = \lambda B_{11} u_1$. The eigenvalues and eigenvectors of the GEVP correspond to eigenvalues and eigenvectors of the matrix $(A_{11} - A_{12} A_{22}^{-1} A_{21}) B_{11}^{-1}$$B_{11}^{-1}(A_{11} - A_{12} A_{22}^{-1} A_{21})$.

Let's write this in block matrices: $$B = \pmatrix{B_{11} & 0\cr 0 & 0\cr}, \ A = \pmatrix{A_{11} & A_{12}\cr A_{21} & A_{22}\cr}, u = \pmatrix{u_1 \cr u_2\cr}$$ where $B_{11}$ has full rank. Then the eigenvector equations $A u = \lambda B u$ become $A_{11} u_1 + A_{12} u_2 = \lambda B_{11} u_1$ and $A_{21} u_1 + A_{22} u_2 = 0$. Suppose $A_{22}$ is invertible. Then we have $u_2 = - A_{22}^{-1} A_{21} u_1$, and $(A_{11} - A_{12} A_{22}^{-1} A_{21}) u_1 = \lambda B_{11} u_1$. The eigenvalues and eigenvectors of the GEVP correspond to eigenvalues and eigenvectors of the matrix $(A_{11} - A_{12} A_{22}^{-1} A_{21}) B_{11}^{-1}$.

Let's write this in block matrices: $$B = \pmatrix{B_{11} & 0\cr 0 & 0\cr}, \ A = \pmatrix{A_{11} & A_{12}\cr A_{21} & A_{22}\cr}, u = \pmatrix{u_1 \cr u_2\cr}$$ where $B_{11}$ has full rank. Then the eigenvector equations $A u = \lambda B u$ become $A_{11} u_1 + A_{12} u_2 = \lambda B_{11} u_1$ and $A_{21} u_1 + A_{22} u_2 = 0$. Suppose $A_{22}$ is invertible. Then we have $u_2 = - A_{22}^{-1} A_{21} u_1$, and $(A_{11} - A_{12} A_{22}^{-1} A_{21}) u_1 = \lambda B_{11} u_1$. The eigenvalues and eigenvectors of the GEVP correspond to eigenvalues and eigenvectors of the matrix $B_{11}^{-1}(A_{11} - A_{12} A_{22}^{-1} A_{21})$.

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Robert Israel
Robert Israel
  • 54.2k
  • 1
  • 76
  • 152

Let's write this in block matrices: $$B = \pmatrix{B_{11} & 0\cr 0 & 0\cr}, \ A = \pmatrix{A_{11} & A_{12}\cr A_{21} & A_{22}\cr}, u = \pmatrix{u_1 \cr u_2\cr}$$ where $B_{11}$ has full rank. Then the eigenvector equations $A u = \lambda B u$ become $A_{11} u_1 + A_{12} u_2 = \lambda B_{11} u_1$ and $A_{21} u_1 + A_{22} u_2 = 0$. Suppose $A_{22}$ is invertible. Then we have $u_2 = - A_{22}^{-1} A_{21} u_1$, and $(A_{11} - A_{12} A_{22}^{-1} A_{21}) u_1 = \lambda B_{11} u_1$. The eigenvalues and eigenvectors of the GEVP correspond to eigenvalues and eigenvectors of the matrix $(A_{11} - A_{12} A_{22}^{-1} A_{21}) B_{11}^{-1}$.