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Weeson Dorne
Weeson Dorne
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A, B$A, B$ are two symmetric matrices, if A-B$ A-B $ is semidefinite (i.e. A - B \geq 0$ A - B \geq 0$), if we rearrange the eigenvalues of two matrices, \lambda_1 (A) \geq \lambda_2 (A) \geq ... \geq \lambda_n (A)$\lambda_1 (A) \geq \lambda_2 (A) \geq ... \geq \lambda_n (A)$, and the same for B$ B $, can we say \lambda_i (A) \geq \lambda_i (B)$ \lambda_i (A) \geq \lambda_i (B) $ for each i$i$ ?

A, B are two symmetric matrices, if A-B is semidefinite (i.e. A - B \geq 0), if we rearrange the eigenvalues of two matrices, \lambda_1 (A) \geq \lambda_2 (A) \geq ... \geq \lambda_n (A), and the same for B, can we say \lambda_i (A) \geq \lambda_i (B) for each i ?

$A, B$ are two symmetric matrices, if $ A-B $ is semidefinite (i.e.$ A - B \geq 0$), if we rearrange the eigenvalues of two matrices, $\lambda_1 (A) \geq \lambda_2 (A) \geq ... \geq \lambda_n (A)$, and the same for $ B $, can we say $ \lambda_i (A) \geq \lambda_i (B) $ for each $i$ ?

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Weeson Dorne
Weeson Dorne
  • 25
  • 1
  • 8

A - B is semidefinite, what the relationship about their eigenvalues?

A, B are two symmetric matrices, if A-B is semidefinite (i.e. A - B \geq 0), if we rearrange the eigenvalues of two matrices, \lambda_1 (A) \geq \lambda_2 (A) \geq ... \geq \lambda_n (A), and the same for B, can we say \lambda_i (A) \geq \lambda_i (B) for each i ?