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m Explain how to compute w with the value of beta and z |
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:<math>T = \begin{bmatrix} Q_{1} & \\ & Q_{2} \end{bmatrix} \left( \begin{bmatrix} D_{1} & \\ & D_{2} \end{bmatrix} + \beta z z^{T} \right) \begin{bmatrix} Q_{1}^{T} & \\ & Q_{2}^{T} \end{bmatrix}</math>
The remaining task has been reduced to finding the eigenvalues of a diagonal matrix plus a rank-one correction. Before showing how to do this, let us simplify the notation. We are looking for the eigenvalues of the matrix <math>D + w w^{T}</math>, where <math>D</math> is diagonal with distinct entries and <math>w</math> is any vector with nonzero entries. In this case <math>w = \sqrt{|\beta|}\cdot z</math>.
The case of a zero entry is simple, since if w<sub>i</sub> is zero, (<math>e_i</math>,d<sub>i</sub>) is an eigenpair (<math>e_i</math> is in the standard basis) of <math>D + w w^{T}</math> since
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