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To compute the eigenvector corresponding to a dominant eigenvalue of a matrix $A\in\mathbb{R}^{n\times n}$, one could apply the Power Iteration: $$v_1=\frac{Av_1}{\|Av_1\|}.$$

  1. in case $A$ is symmetric, eigenvectors are orthonormal. However, suppose that there are, e.g, two occurrences of the dominant eigenvalue $\lambda_1$ corresponding to different eigenvectors. Does that mean that the method would yield inconsistent results on different invocation? The "inconsistency" means that the method could (assuming random initialization on each invocation) change direction of convergence (since the eigenvalues are the same)

In case one needs the following dominant eigenvector, one usually performs Gram Schmidt orthonormalization, ie, removes component of the first eigenvector from the initialization to the second. Would this second vector converge to the eigenvector corresponding to "the other" occurence of dominanant eigenvalue $\lambda_1$?

  1. in case of a general $A$, eigenvectors are not orthonormal. So, what would be the way to extract subsequent eigenvectors. In other words, would GS orthonormalization now make sense? It removes component from the first eigenvector, but, since the eigenvectors are not orthogonal, I'm not sure it the following matrix-vector multiplication adds the component back.

To compute the eigenvector corresponding to a dominant eigenvalue of a matrix $A\in\mathbb{R}^{n\times n}$, one could apply the Power Iteration: $$v_1=\frac{Av_1}{\|Av_1\|}.$$

  1. in case $A$ is symmetric, eigenvectors are orthonormal. However, suppose that there are, e.g, two occurrences of the dominant eigenvalue $\lambda_1$ corresponding to different eigenvectors. Does that mean that the method would yield inconsistent results on different invocation?

In case one needs the following dominant eigenvector, one usually performs Gram Schmidt orthonormalization, ie, removes component of the first eigenvector from the initialization to the second. Would this second vector converge to the eigenvector corresponding to "the other" occurence of dominanant eigenvalue $\lambda_1$?

  1. in case of a general $A$, eigenvectors are not orthonormal. So, what would be the way to extract subsequent eigenvectors. In other words, would GS orthonormalization now make sense? It removes component from the first eigenvector, but, since the eigenvectors are not orthogonal, I'm not sure it the following matrix-vector multiplication adds the component back.

To compute the eigenvector corresponding to a dominant eigenvalue of a matrix $A\in\mathbb{R}^{n\times n}$, one could apply the Power Iteration: $$v_1=\frac{Av_1}{\|Av_1\|}.$$

  1. in case $A$ is symmetric, eigenvectors are orthonormal. However, suppose that there are, e.g, two occurrences of the dominant eigenvalue $\lambda_1$ corresponding to different eigenvectors. Does that mean that the method would yield inconsistent results on different invocation? The "inconsistency" means that the method could (assuming random initialization on each invocation) change direction of convergence (since the eigenvalues are the same)

In case one needs the following dominant eigenvector, one usually performs Gram Schmidt orthonormalization, ie, removes component of the first eigenvector from the initialization to the second. Would this second vector converge to the eigenvector corresponding to "the other" occurence of dominanant eigenvalue $\lambda_1$?

  1. in case of a general $A$, eigenvectors are not orthonormal. So, what would be the way to extract subsequent eigenvectors. In other words, would GS orthonormalization now make sense? It removes component from the first eigenvector, but, since the eigenvectors are not orthogonal, I'm not sure it the following matrix-vector multiplication adds the component back.
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usero
usero
  • 1.7k
  • 2
  • 14
  • 27

Power Iteration on general matrices (with higher multiplicity of dominant eigenvalue)

To compute the eigenvector corresponding to a dominant eigenvalue of a matrix $A\in\mathbb{R}^{n\times n}$, one could apply the Power Iteration: $$v_1=\frac{Av_1}{\|Av_1\|}.$$

  1. in case $A$ is symmetric, eigenvectors are orthonormal. However, suppose that there are, e.g, two occurrences of the dominant eigenvalue $\lambda_1$ corresponding to different eigenvectors. Does that mean that the method would yield inconsistent results on different invocation?

In case one needs the following dominant eigenvector, one usually performs Gram Schmidt orthonormalization, ie, removes component of the first eigenvector from the initialization to the second. Would this second vector converge to the eigenvector corresponding to "the other" occurence of dominanant eigenvalue $\lambda_1$?

  1. in case of a general $A$, eigenvectors are not orthonormal. So, what would be the way to extract subsequent eigenvectors. In other words, would GS orthonormalization now make sense? It removes component from the first eigenvector, but, since the eigenvectors are not orthogonal, I'm not sure it the following matrix-vector multiplication adds the component back.