Removing a sparsity constraint in operator bound The Next CEO of Stack OverflowFETI domain decomposition - kernel of local stiffness matricesWhat are the conditions should be added so that submatrix has full rankProbablistic bound for $|RR^TM|$ for uniformly random orthonormal matrix $R$Largest eigenvalue of a Hermitian matrix$Ax = kx$ for complex scalar $k$ and complex vector $x$, and symmetric matrix $A$In a binary code, all coordinates partake in at least one non-information setIf $A$ is of rank $n$ then why is it non-singular?How close are solutions of systems of homogeneous linear inequalities with close coefficients?Show that if $Av_1,ldots,Av_n$ is linearly independent then $A$ is non-singularThe number of non-singular $ntimes n$ matrices over $mathbbF_2$ with exactly $k$ non-zero entries
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Removing a sparsity constraint in operator bound
The Next CEO of Stack OverflowFETI domain decomposition - kernel of local stiffness matricesWhat are the conditions should be added so that submatrix has full rankProbablistic bound for $|RR^TM|$ for uniformly random orthonormal matrix $R$Largest eigenvalue of a Hermitian matrix$Ax = kx$ for complex scalar $k$ and complex vector $x$, and symmetric matrix $A$In a binary code, all coordinates partake in at least one non-information setIf $A$ is of rank $n$ then why is it non-singular?How close are solutions of systems of homogeneous linear inequalities with close coefficients?Show that if $Av_1,ldots,Av_n$ is linearly independent then $A$ is non-singularThe number of non-singular $ntimes n$ matrices over $mathbbF_2$ with exactly $k$ non-zero entries
$begingroup$
Suppose that $A in mathbfR^n times n$ satisfies
$$
sqrt1 - delta |x|_2 leq |Ax|_2 leq sqrt1 + delta |x|_2
$$
for all $x in mathbfR^n$ with exactly $s$ non-zero coordinates.
Apparently, then for any $x in mathbfR^n$,
$$
|Ax|_2 leq sqrt1 + deltaleft(|x|_2 + fracxsqrtsright).
$$
I tried showing this by splitting the support of $x$ into disjoint sets, but couldn't match the second term. Can anyone help?
linear-algebra
asked Mar 28 at 0:36
Drew BradyDrew Brady
731315
$endgroup$
add a comment |
$begingroup$
Suppose that $A in mathbfR^n times n$ satisfies
$$
sqrt1 - delta |x|_2 leq |Ax|_2 leq sqrt1 + delta |x|_2
$$
for all $x in mathbfR^n$ with exactly $s$ non-zero coordinates.
Apparently, then for any $x in mathbfR^n$,
$$
|Ax|_2 leq sqrt1 + deltaleft(|x|_2 + fracxsqrtsright).
$$
I tried showing this by splitting the support of $x$ into disjoint sets, but couldn't match the second term. Can anyone help?
linear-algebra
asked Mar 28 at 0:36
Drew BradyDrew Brady
731315
$endgroup$
$begingroup$
What is $sqrt s$?
$endgroup$
– xpaul
Mar 28 at 2:32
$begingroup$
@xpaul as stated above, the first pair of inequalities hold for all $x$ with exactly $s$ non-zeros.
$endgroup$
– Drew Brady
Mar 28 at 4:08
add a comment |
$begingroup$
Suppose that $A in mathbfR^n times n$ satisfies
$$
sqrt1 - delta |x|_2 leq |Ax|_2 leq sqrt1 + delta |x|_2
$$
for all $x in mathbfR^n$ with exactly $s$ non-zero coordinates.
Apparently, then for any $x in mathbfR^n$,
$$
|Ax|_2 leq sqrt1 + deltaleft(|x|_2 + fracxsqrtsright).
$$
I tried showing this by splitting the support of $x$ into disjoint sets, but couldn't match the second term. Can anyone help?
linear-algebra
asked Mar 28 at 0:36
Drew BradyDrew Brady
731315
$endgroup$
Suppose that $A in mathbfR^n times n$ satisfies
$$
sqrt1 - delta |x|_2 leq |Ax|_2 leq sqrt1 + delta |x|_2
$$
for all $x in mathbfR^n$ with exactly $s$ non-zero coordinates.
Apparently, then for any $x in mathbfR^n$,
$$
|Ax|_2 leq sqrt1 + deltaleft(|x|_2 + fracxsqrtsright).
$$
I tried showing this by splitting the support of $x$ into disjoint sets, but couldn't match the second term. Can anyone help?
linear-algebra
linear-algebra
asked Mar 28 at 0:36
Drew BradyDrew Brady
731315
asked Mar 28 at 0:36
Drew BradyDrew Brady
731315
asked Mar 28 at 0:36
Drew BradyDrew Brady
731315
asked Mar 28 at 0:36
Drew BradyDrew Brady
731315
asked Mar 28 at 0:36
Drew BradyDrew Brady
731315
731315
$begingroup$
What is $sqrt s$?
$endgroup$
– xpaul
Mar 28 at 2:32
$begingroup$
@xpaul as stated above, the first pair of inequalities hold for all $x$ with exactly $s$ non-zeros.
$endgroup$
– Drew Brady
Mar 28 at 4:08
add a comment |
$begingroup$
What is $sqrt s$?
$endgroup$
– xpaul
Mar 28 at 2:32
$begingroup$
@xpaul as stated above, the first pair of inequalities hold for all $x$ with exactly $s$ non-zeros.
$endgroup$
– Drew Brady
Mar 28 at 4:08
$begingroup$
What is $sqrt s$?
$endgroup$
– xpaul
Mar 28 at 2:32
$begingroup$
What is $sqrt s$?
$endgroup$
– xpaul
Mar 28 at 2:32
$begingroup$
@xpaul as stated above, the first pair of inequalities hold for all $x$ with exactly $s$ non-zeros.
$endgroup$
– Drew Brady
Mar 28 at 4:08
$begingroup$
@xpaul as stated above, the first pair of inequalities hold for all $x$ with exactly $s$ non-zeros.
$endgroup$
– Drew Brady
Mar 28 at 4:08
add a comment |
0
active
oldest
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$begingroup$
What is $sqrt s$?
$endgroup$
– xpaul
Mar 28 at 2:32
$begingroup$
@xpaul as stated above, the first pair of inequalities hold for all $x$ with exactly $s$ non-zeros.
$endgroup$
– Drew Brady
Mar 28 at 4:08