Hadamard product: Optimal bound on operator norm
Let $A,B$ be $ntimes n $ matrices and denote by $Astar B$ the Hadamard product $(Astar B)(i,j)=A(i,j)B(i,j)$ (pointwise matrix multiplication). For $A$ positive definite it is known that
$$|Astar B| leq sup_{i,j} |A(i,j)| |B|.$$
My question is what happens if we drop the positive definiteness assumption, i.e. what is the best constant $C>0$ such that
$$|Astar B| leq C sup_{i,j} |A(i,j)| |B|$$
holds for arbitrary $ntimes n$ matrices $A,B$. Is the constant $C$ independent of the size of the matrix $n$?
linear-algebra matrices functional-analysis
asked Sep 6 '18 at 21:14
SanneSanne
485
add a comment |
Let $A,B$ be $ntimes n $ matrices and denote by $Astar B$ the Hadamard product $(Astar B)(i,j)=A(i,j)B(i,j)$ (pointwise matrix multiplication). For $A$ positive definite it is known that
$$|Astar B| leq sup_{i,j} |A(i,j)| |B|.$$
My question is what happens if we drop the positive definiteness assumption, i.e. what is the best constant $C>0$ such that
$$|Astar B| leq C sup_{i,j} |A(i,j)| |B|$$
holds for arbitrary $ntimes n$ matrices $A,B$. Is the constant $C$ independent of the size of the matrix $n$?
linear-algebra matrices functional-analysis
asked Sep 6 '18 at 21:14
SanneSanne
485
add a comment |
Let $A,B$ be $ntimes n $ matrices and denote by $Astar B$ the Hadamard product $(Astar B)(i,j)=A(i,j)B(i,j)$ (pointwise matrix multiplication). For $A$ positive definite it is known that
$$|Astar B| leq sup_{i,j} |A(i,j)| |B|.$$
My question is what happens if we drop the positive definiteness assumption, i.e. what is the best constant $C>0$ such that
$$|Astar B| leq C sup_{i,j} |A(i,j)| |B|$$
holds for arbitrary $ntimes n$ matrices $A,B$. Is the constant $C$ independent of the size of the matrix $n$?
linear-algebra matrices functional-analysis
asked Sep 6 '18 at 21:14
SanneSanne
485
Let $A,B$ be $ntimes n $ matrices and denote by $Astar B$ the Hadamard product $(Astar B)(i,j)=A(i,j)B(i,j)$ (pointwise matrix multiplication). For $A$ positive definite it is known that
$$|Astar B| leq sup_{i,j} |A(i,j)| |B|.$$
My question is what happens if we drop the positive definiteness assumption, i.e. what is the best constant $C>0$ such that
$$|Astar B| leq C sup_{i,j} |A(i,j)| |B|$$
holds for arbitrary $ntimes n$ matrices $A,B$. Is the constant $C$ independent of the size of the matrix $n$?
linear-algebra matrices functional-analysis
linear-algebra matrices functional-analysis
asked Sep 6 '18 at 21:14
SanneSanne
485
asked Sep 6 '18 at 21:14
SanneSanne
485
asked Sep 6 '18 at 21:14
SanneSanne
485
asked Sep 6 '18 at 21:14
SanneSanne
485
asked Sep 6 '18 at 21:14
SanneSanne
485
485
add a comment |
add a comment |
1 Answer
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The optimal such $C$ is $sqrt{n}$, and so in particular there does not exist any such $C$ that is independent of $n$.
First, let me prove that $C=sqrt{n}$ works. Let $e_1,dots,e_n$ be the standard basis vectors, let $v=sum v_ie_i$ be any vector, and let $a=sup_{i,j}|A(i,j)|$. Note that for each $i$, $|(Astar B)e_i|leq a|B|$, since $|Be_i|leq |B|$ and $(Astar B)e_i$ is obtained from $Be_i$ by multiplying each entry by a scalar of size at most $a$. Thus $$|(Astar B)v|leqsum|v_i||(Astar B)e_i|leq a|B|sum|v_i|.$$ By Cauchy-Schwarz, $sum|v_i|leq sqrt{n}|v|$, so we conclude that $|Astar B|leq a|B|sqrt{n}$ and $C=sqrt{n}$ works.
To prove $C=sqrt{n}$ is optimal, let $omega$ be a primitive $n$th root of unity and let $B(i,j)=frac{omega^{ij}}{sqrt{n}}$. The columns of $B$ are orthonormal, so $B$ is unitary and $|B|=1$. Now let $A(i,j)=omega^{-ij}$, so that $Astar B$ is the matrix whose entries are all $frac{1}{sqrt{n}}$. We have $|Astar B|=sqrt{n}$ (the vector of all $1$s is an eigenvector of $Astar B$ with eigenvalue $sqrt{n}$). Since $|A(i,j)|=1$ for all $i,j$, this means we must have $Cgeq sqrt{n}$.
answered Sep 6 '18 at 22:24
Eric WofseyEric Wofsey
180k12207335
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The optimal such $C$ is $sqrt{n}$, and so in particular there does not exist any such $C$ that is independent of $n$.
First, let me prove that $C=sqrt{n}$ works. Let $e_1,dots,e_n$ be the standard basis vectors, let $v=sum v_ie_i$ be any vector, and let $a=sup_{i,j}|A(i,j)|$. Note that for each $i$, $|(Astar B)e_i|leq a|B|$, since $|Be_i|leq |B|$ and $(Astar B)e_i$ is obtained from $Be_i$ by multiplying each entry by a scalar of size at most $a$. Thus $$|(Astar B)v|leqsum|v_i||(Astar B)e_i|leq a|B|sum|v_i|.$$ By Cauchy-Schwarz, $sum|v_i|leq sqrt{n}|v|$, so we conclude that $|Astar B|leq a|B|sqrt{n}$ and $C=sqrt{n}$ works.
To prove $C=sqrt{n}$ is optimal, let $omega$ be a primitive $n$th root of unity and let $B(i,j)=frac{omega^{ij}}{sqrt{n}}$. The columns of $B$ are orthonormal, so $B$ is unitary and $|B|=1$. Now let $A(i,j)=omega^{-ij}$, so that $Astar B$ is the matrix whose entries are all $frac{1}{sqrt{n}}$. We have $|Astar B|=sqrt{n}$ (the vector of all $1$s is an eigenvector of $Astar B$ with eigenvalue $sqrt{n}$). Since $|A(i,j)|=1$ for all $i,j$, this means we must have $Cgeq sqrt{n}$.
answered Sep 6 '18 at 22:24
Eric WofseyEric Wofsey
180k12207335
add a comment |
The optimal such $C$ is $sqrt{n}$, and so in particular there does not exist any such $C$ that is independent of $n$.
First, let me prove that $C=sqrt{n}$ works. Let $e_1,dots,e_n$ be the standard basis vectors, let $v=sum v_ie_i$ be any vector, and let $a=sup_{i,j}|A(i,j)|$. Note that for each $i$, $|(Astar B)e_i|leq a|B|$, since $|Be_i|leq |B|$ and $(Astar B)e_i$ is obtained from $Be_i$ by multiplying each entry by a scalar of size at most $a$. Thus $$|(Astar B)v|leqsum|v_i||(Astar B)e_i|leq a|B|sum|v_i|.$$ By Cauchy-Schwarz, $sum|v_i|leq sqrt{n}|v|$, so we conclude that $|Astar B|leq a|B|sqrt{n}$ and $C=sqrt{n}$ works.
To prove $C=sqrt{n}$ is optimal, let $omega$ be a primitive $n$th root of unity and let $B(i,j)=frac{omega^{ij}}{sqrt{n}}$. The columns of $B$ are orthonormal, so $B$ is unitary and $|B|=1$. Now let $A(i,j)=omega^{-ij}$, so that $Astar B$ is the matrix whose entries are all $frac{1}{sqrt{n}}$. We have $|Astar B|=sqrt{n}$ (the vector of all $1$s is an eigenvector of $Astar B$ with eigenvalue $sqrt{n}$). Since $|A(i,j)|=1$ for all $i,j$, this means we must have $Cgeq sqrt{n}$.
answered Sep 6 '18 at 22:24
Eric WofseyEric Wofsey
180k12207335
add a comment |
The optimal such $C$ is $sqrt{n}$, and so in particular there does not exist any such $C$ that is independent of $n$.
First, let me prove that $C=sqrt{n}$ works. Let $e_1,dots,e_n$ be the standard basis vectors, let $v=sum v_ie_i$ be any vector, and let $a=sup_{i,j}|A(i,j)|$. Note that for each $i$, $|(Astar B)e_i|leq a|B|$, since $|Be_i|leq |B|$ and $(Astar B)e_i$ is obtained from $Be_i$ by multiplying each entry by a scalar of size at most $a$. Thus $$|(Astar B)v|leqsum|v_i||(Astar B)e_i|leq a|B|sum|v_i|.$$ By Cauchy-Schwarz, $sum|v_i|leq sqrt{n}|v|$, so we conclude that $|Astar B|leq a|B|sqrt{n}$ and $C=sqrt{n}$ works.
To prove $C=sqrt{n}$ is optimal, let $omega$ be a primitive $n$th root of unity and let $B(i,j)=frac{omega^{ij}}{sqrt{n}}$. The columns of $B$ are orthonormal, so $B$ is unitary and $|B|=1$. Now let $A(i,j)=omega^{-ij}$, so that $Astar B$ is the matrix whose entries are all $frac{1}{sqrt{n}}$. We have $|Astar B|=sqrt{n}$ (the vector of all $1$s is an eigenvector of $Astar B$ with eigenvalue $sqrt{n}$). Since $|A(i,j)|=1$ for all $i,j$, this means we must have $Cgeq sqrt{n}$.
answered Sep 6 '18 at 22:24
Eric WofseyEric Wofsey
180k12207335
The optimal such $C$ is $sqrt{n}$, and so in particular there does not exist any such $C$ that is independent of $n$.
First, let me prove that $C=sqrt{n}$ works. Let $e_1,dots,e_n$ be the standard basis vectors, let $v=sum v_ie_i$ be any vector, and let $a=sup_{i,j}|A(i,j)|$. Note that for each $i$, $|(Astar B)e_i|leq a|B|$, since $|Be_i|leq |B|$ and $(Astar B)e_i$ is obtained from $Be_i$ by multiplying each entry by a scalar of size at most $a$. Thus $$|(Astar B)v|leqsum|v_i||(Astar B)e_i|leq a|B|sum|v_i|.$$ By Cauchy-Schwarz, $sum|v_i|leq sqrt{n}|v|$, so we conclude that $|Astar B|leq a|B|sqrt{n}$ and $C=sqrt{n}$ works.
To prove $C=sqrt{n}$ is optimal, let $omega$ be a primitive $n$th root of unity and let $B(i,j)=frac{omega^{ij}}{sqrt{n}}$. The columns of $B$ are orthonormal, so $B$ is unitary and $|B|=1$. Now let $A(i,j)=omega^{-ij}$, so that $Astar B$ is the matrix whose entries are all $frac{1}{sqrt{n}}$. We have $|Astar B|=sqrt{n}$ (the vector of all $1$s is an eigenvector of $Astar B$ with eigenvalue $sqrt{n}$). Since $|A(i,j)|=1$ for all $i,j$, this means we must have $Cgeq sqrt{n}$.
answered Sep 6 '18 at 22:24
Eric WofseyEric Wofsey
180k12207335
edited Jan 4 at 6:11
answered Sep 6 '18 at 22:24
Eric WofseyEric Wofsey
180k12207335
answered Sep 6 '18 at 22:24
Eric WofseyEric Wofsey
180k12207335
answered Sep 6 '18 at 22:24
Eric WofseyEric Wofsey
180k12207335
180k12207335
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