Upper bound of spectral radius of the sum of two matrices, one with spectral radius no larger than 1, and the other has small eigenvalues

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Suppose I have one $pNtimes pN$ matrix $bf A$ with spectral radius no larger than 1 (maximum of absolute values of eigenvalues is no larger than 1), and the other matrix $bf H$ is in a block-like format (empty means zero, only zeros in the top-left and bottom-right block are explicitly marked, the superscript like $N^(N+1,N)$ means this number "$N$" is at the $N+1$th row and $N$th column)



enter image description here



My question is how to derive a reasonably tight bound of the spectral radius of the sum $bf A+H$. Again the spectral radius of $bf A$ is smaller than 1. The eigenvalue of $bf H$ is $pm frac1n + 1$, so we believe the spectral radius of $bfA+H$ should be near the spectral radius of $bf A$.




linear-algebra eigenvalues-eigenvectors




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  • @user1551 Sorry the H was in wrong format and I missed the coefficient. Thanks!
    – Tony
    Aug 28 at 0:46














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Suppose I have one $pNtimes pN$ matrix $bf A$ with spectral radius no larger than 1 (maximum of absolute values of eigenvalues is no larger than 1), and the other matrix $bf H$ is in a block-like format (empty means zero, only zeros in the top-left and bottom-right block are explicitly marked, the superscript like $N^(N+1,N)$ means this number "$N$" is at the $N+1$th row and $N$th column)



enter image description here



My question is how to derive a reasonably tight bound of the spectral radius of the sum $bf A+H$. Again the spectral radius of $bf A$ is smaller than 1. The eigenvalue of $bf H$ is $pm frac1n + 1$, so we believe the spectral radius of $bfA+H$ should be near the spectral radius of $bf A$.




linear-algebra eigenvalues-eigenvectors




share|cite|improve this question






















  • @user1551 Sorry the H was in wrong format and I missed the coefficient. Thanks!
    – Tony
    Aug 28 at 0:46












up vote
3
down vote

favorite









up vote
3
down vote

favorite











Suppose I have one $pNtimes pN$ matrix $bf A$ with spectral radius no larger than 1 (maximum of absolute values of eigenvalues is no larger than 1), and the other matrix $bf H$ is in a block-like format (empty means zero, only zeros in the top-left and bottom-right block are explicitly marked, the superscript like $N^(N+1,N)$ means this number "$N$" is at the $N+1$th row and $N$th column)



enter image description here



My question is how to derive a reasonably tight bound of the spectral radius of the sum $bf A+H$. Again the spectral radius of $bf A$ is smaller than 1. The eigenvalue of $bf H$ is $pm frac1n + 1$, so we believe the spectral radius of $bfA+H$ should be near the spectral radius of $bf A$.




linear-algebra eigenvalues-eigenvectors




share|cite|improve this question














Suppose I have one $pNtimes pN$ matrix $bf A$ with spectral radius no larger than 1 (maximum of absolute values of eigenvalues is no larger than 1), and the other matrix $bf H$ is in a block-like format (empty means zero, only zeros in the top-left and bottom-right block are explicitly marked, the superscript like $N^(N+1,N)$ means this number "$N$" is at the $N+1$th row and $N$th column)



enter image description here



My question is how to derive a reasonably tight bound of the spectral radius of the sum $bf A+H$. Again the spectral radius of $bf A$ is smaller than 1. The eigenvalue of $bf H$ is $pm frac1n + 1$, so we believe the spectral radius of $bfA+H$ should be near the spectral radius of $bf A$.





linear-algebra eigenvalues-eigenvectors





share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 28 at 1:31

























asked Aug 27 at 18:06









Tony

2,1491626




2,1491626











  • @user1551 Sorry the H was in wrong format and I missed the coefficient. Thanks!
    – Tony
    Aug 28 at 0:46
















  • @user1551 Sorry the H was in wrong format and I missed the coefficient. Thanks!
    – Tony
    Aug 28 at 0:46















@user1551 Sorry the H was in wrong format and I missed the coefficient. Thanks!
– Tony
Aug 28 at 0:46




@user1551 Sorry the H was in wrong format and I missed the coefficient. Thanks!
– Tony
Aug 28 at 0:46















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