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PThomasCS
PThomasCS
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I have that a sequence of random matrices, $M_n$, converges almost surely to a diagonal matrix, $D$, with finite real entries on its diagonal. During convergence, the off-diagonals are not necessarily zero. How can I go about showing that the smallest eigenvector of $M_n$ converges almost surely to the smallest eigenvector of $D$?

Edit: If it helps, at least one diagonal entry of $D$ is zero and each $M_n$ is symmetric.

I have that a sequence of random matrices, $M_n$, converges almost surely to a diagonal matrix, $D$, with finite real entries on its diagonal. During convergence, the off-diagonals are not necessarily zero. How can I go about showing that the smallest eigenvector of $M_n$ converges almost surely to the smallest eigenvector of $D$?

Edit: If it helps, at least one diagonal entry of $D$ is zero.

I have that a sequence of random matrices, $M_n$, converges almost surely to a diagonal matrix, $D$, with finite real entries on its diagonal. During convergence, the off-diagonals are not necessarily zero. How can I go about showing that the smallest eigenvector of $M_n$ converges almost surely to the smallest eigenvector of $D$?

Edit: If it helps, at least one diagonal entry of $D$ is zero and each $M_n$ is symmetric.

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PThomasCS
PThomasCS
  • 399
  • 1
  • 9

I have that a sequence of random matrices, $M_n$, converges almost surely to a diagonal matrix, $D$, with finite real entries on its diagonal. During convergence, the off-diagonals are not necessarily zero. How can I go about showing that the smallest eigenvector of $M_n$ converges almost surely to the smallest eigenvector of $D$?

Edit: If it helps, at least one diagonal entry of $D$ is zero.

I have that a sequence of random matrices, $M_n$, converges almost surely to a diagonal matrix, $D$, with finite real entries on its diagonal. During convergence, the off-diagonals are not necessarily zero. How can I go about showing that the smallest eigenvector of $M_n$ converges almost surely to the smallest eigenvector of $D$?

I have that a sequence of random matrices, $M_n$, converges almost surely to a diagonal matrix, $D$, with finite real entries on its diagonal. During convergence, the off-diagonals are not necessarily zero. How can I go about showing that the smallest eigenvector of $M_n$ converges almost surely to the smallest eigenvector of $D$?

Edit: If it helps, at least one diagonal entry of $D$ is zero.

Source Link
PThomasCS
PThomasCS
  • 399
  • 1
  • 9

Almost sure convergence of smallest eigenvector of diagonal matrix

I have that a sequence of random matrices, $M_n$, converges almost surely to a diagonal matrix, $D$, with finite real entries on its diagonal. During convergence, the off-diagonals are not necessarily zero. How can I go about showing that the smallest eigenvector of $M_n$ converges almost surely to the smallest eigenvector of $D$?