Matrix condition number and reorderingprecision vs matrix condition numberFastest algorithm to compute the...
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Matrix condition number and reordering
precision vs matrix condition numberFastest algorithm to compute the condition number of a large matrix in Matlab/OctaveHow to approximate the condition number of a large matrix?Condition number of $X^{T}AX$Applying the result of Cuthill-McKee in SciPyIs large condition number good measure of nearness to singularity for a matrix?Condition number of matrix and effects of round off errorsCondition number of two perburbation matrix regarding limit and quadtrature integration rulesCondition number of a matrixCHOLMOD condition number estimate
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Does the condition number change when a matrix is reordered by e.g. Cuthill Mckee or some other method?
linear-algebra matrix condition-number
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Does the condition number change when a matrix is reordered by e.g. Cuthill Mckee or some other method?
linear-algebra matrix condition-number
asked 8 hours ago
vydesastervydesaster
1137 bronze badges
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add a comment |
$begingroup$
Does the condition number change when a matrix is reordered by e.g. Cuthill Mckee or some other method?
linear-algebra matrix condition-number
asked 8 hours ago
vydesastervydesaster
1137 bronze badges
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Does the condition number change when a matrix is reordered by e.g. Cuthill Mckee or some other method?
linear-algebra matrix condition-number
linear-algebra matrix condition-number
asked 8 hours ago
vydesastervydesaster
1137 bronze badges
asked 8 hours ago
vydesastervydesaster
1137 bronze badges
asked 8 hours ago
vydesastervydesaster
1137 bronze badges
asked 8 hours ago
vydesastervydesaster
1137 bronze badges
asked 8 hours ago
vydesastervydesaster
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1137 bronze badges
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No. A reordering of a matrix $boldsymbol{A}$ is equivalent to conjugation by a permutation matrix $boldsymbol{P}$. In other words, the reordered matrix can be written as $boldsymbol{A}_r = boldsymbol{P}boldsymbol{A}boldsymbol{P}^*$ and
$boldsymbol{A}_r^{-1} = boldsymbol{P}boldsymbol{A}^{-1}boldsymbol{P}^*$.
Since the spectral norm is unitarily invariant and permutation matrices are unitary, the $ell^2$ condition number $$kappa(boldsymbol{A}_r) = |boldsymbol{A}_r|_2|boldsymbol{A}_r^{-1}|_2 = |boldsymbol{P}boldsymbol{A}boldsymbol{P}^*|_2|boldsymbol{P}boldsymbol{A}^{-1}boldsymbol{P}^*|_2 = |boldsymbol{A}|_2|boldsymbol{A}^{-1}|_2 = kappa(boldsymbol{A}).$$
answered 7 hours ago
cdipaolocdipaolo
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1 Answer
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1 Answer
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$begingroup$
No. A reordering of a matrix $boldsymbol{A}$ is equivalent to conjugation by a permutation matrix $boldsymbol{P}$. In other words, the reordered matrix can be written as $boldsymbol{A}_r = boldsymbol{P}boldsymbol{A}boldsymbol{P}^*$ and
$boldsymbol{A}_r^{-1} = boldsymbol{P}boldsymbol{A}^{-1}boldsymbol{P}^*$.
Since the spectral norm is unitarily invariant and permutation matrices are unitary, the $ell^2$ condition number $$kappa(boldsymbol{A}_r) = |boldsymbol{A}_r|_2|boldsymbol{A}_r^{-1}|_2 = |boldsymbol{P}boldsymbol{A}boldsymbol{P}^*|_2|boldsymbol{P}boldsymbol{A}^{-1}boldsymbol{P}^*|_2 = |boldsymbol{A}|_2|boldsymbol{A}^{-1}|_2 = kappa(boldsymbol{A}).$$
answered 7 hours ago
cdipaolocdipaolo
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cdipaolo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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add a comment |
$begingroup$
No. A reordering of a matrix $boldsymbol{A}$ is equivalent to conjugation by a permutation matrix $boldsymbol{P}$. In other words, the reordered matrix can be written as $boldsymbol{A}_r = boldsymbol{P}boldsymbol{A}boldsymbol{P}^*$ and
$boldsymbol{A}_r^{-1} = boldsymbol{P}boldsymbol{A}^{-1}boldsymbol{P}^*$.
Since the spectral norm is unitarily invariant and permutation matrices are unitary, the $ell^2$ condition number $$kappa(boldsymbol{A}_r) = |boldsymbol{A}_r|_2|boldsymbol{A}_r^{-1}|_2 = |boldsymbol{P}boldsymbol{A}boldsymbol{P}^*|_2|boldsymbol{P}boldsymbol{A}^{-1}boldsymbol{P}^*|_2 = |boldsymbol{A}|_2|boldsymbol{A}^{-1}|_2 = kappa(boldsymbol{A}).$$
answered 7 hours ago
cdipaolocdipaolo
1463 bronze badges
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cdipaolo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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add a comment |
$begingroup$
No. A reordering of a matrix $boldsymbol{A}$ is equivalent to conjugation by a permutation matrix $boldsymbol{P}$. In other words, the reordered matrix can be written as $boldsymbol{A}_r = boldsymbol{P}boldsymbol{A}boldsymbol{P}^*$ and
$boldsymbol{A}_r^{-1} = boldsymbol{P}boldsymbol{A}^{-1}boldsymbol{P}^*$.
Since the spectral norm is unitarily invariant and permutation matrices are unitary, the $ell^2$ condition number $$kappa(boldsymbol{A}_r) = |boldsymbol{A}_r|_2|boldsymbol{A}_r^{-1}|_2 = |boldsymbol{P}boldsymbol{A}boldsymbol{P}^*|_2|boldsymbol{P}boldsymbol{A}^{-1}boldsymbol{P}^*|_2 = |boldsymbol{A}|_2|boldsymbol{A}^{-1}|_2 = kappa(boldsymbol{A}).$$
answered 7 hours ago
cdipaolocdipaolo
1463 bronze badges
New contributor
cdipaolo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
No. A reordering of a matrix $boldsymbol{A}$ is equivalent to conjugation by a permutation matrix $boldsymbol{P}$. In other words, the reordered matrix can be written as $boldsymbol{A}_r = boldsymbol{P}boldsymbol{A}boldsymbol{P}^*$ and
$boldsymbol{A}_r^{-1} = boldsymbol{P}boldsymbol{A}^{-1}boldsymbol{P}^*$.
Since the spectral norm is unitarily invariant and permutation matrices are unitary, the $ell^2$ condition number $$kappa(boldsymbol{A}_r) = |boldsymbol{A}_r|_2|boldsymbol{A}_r^{-1}|_2 = |boldsymbol{P}boldsymbol{A}boldsymbol{P}^*|_2|boldsymbol{P}boldsymbol{A}^{-1}boldsymbol{P}^*|_2 = |boldsymbol{A}|_2|boldsymbol{A}^{-1}|_2 = kappa(boldsymbol{A}).$$
answered 7 hours ago
cdipaolocdipaolo
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New contributor
cdipaolo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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answered 7 hours ago
cdipaolocdipaolo
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answered 7 hours ago
cdipaolocdipaolo
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answered 7 hours ago
cdipaolocdipaolo
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