Norm of sum of shifted outer products

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Consider two vectors $u, v in mathbbR^m$ satisfying $| u |_2 = | v |_2 = 1$ and $#mathrmsupp(v) leq n < m$, where $mathrmsupp$ denotes the support of the vector (i.e. locations of nonzero coordinates). Additionally, assume that $mathrmsupp(u) = mathrmsupp(v)$. Assume that $u^i$ denotes the left-circular shift of $u$ by $i$ locations, i.e.



$$
u = (u_1, dots, u_n, 0, dots, 0) Rightarrow u^1 =
(0, u_2, dots, u_n, 0 dots, u_1)
$$



Form the following matrix:



$$
X = u v^top + u^1 v^1^top + dots + u^m-1 v^m-1^top
$$




Question:
I want to obtain a nontrivial bound on the operator and Frobenius norms of this matrix. The only bound that I was able to obtain so far is using the subadditivity of the Frobenius norm and the fact that $| u_i v_i^top |_F = 1, ; forall i.$
$$
| X |_mathrmop leq | X |_F leq m | u_i v_i^top | = m
$$




Any ideas or pointers to references are more than welcome.




linear-algebra matrices reference-request spheres matrix-norms




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  • 1




    I'd expect the worst case to be constant $u$ and $v$ on a contiguous support. That would yield a bound of roughly $sqrtnm$ for the Frobenius norm.
    – joriki
    Jul 24 at 4:08














up vote
2
down vote

favorite












Consider two vectors $u, v in mathbbR^m$ satisfying $| u |_2 = | v |_2 = 1$ and $#mathrmsupp(v) leq n < m$, where $mathrmsupp$ denotes the support of the vector (i.e. locations of nonzero coordinates). Additionally, assume that $mathrmsupp(u) = mathrmsupp(v)$. Assume that $u^i$ denotes the left-circular shift of $u$ by $i$ locations, i.e.



$$
u = (u_1, dots, u_n, 0, dots, 0) Rightarrow u^1 =
(0, u_2, dots, u_n, 0 dots, u_1)
$$



Form the following matrix:



$$
X = u v^top + u^1 v^1^top + dots + u^m-1 v^m-1^top
$$




Question:
I want to obtain a nontrivial bound on the operator and Frobenius norms of this matrix. The only bound that I was able to obtain so far is using the subadditivity of the Frobenius norm and the fact that $| u_i v_i^top |_F = 1, ; forall i.$
$$
| X |_mathrmop leq | X |_F leq m | u_i v_i^top | = m
$$




Any ideas or pointers to references are more than welcome.




linear-algebra matrices reference-request spheres matrix-norms




share|cite|improve this question















  • 1




    I'd expect the worst case to be constant $u$ and $v$ on a contiguous support. That would yield a bound of roughly $sqrtnm$ for the Frobenius norm.
    – joriki
    Jul 24 at 4:08












up vote
2
down vote

favorite









up vote
2
down vote

favorite











Consider two vectors $u, v in mathbbR^m$ satisfying $| u |_2 = | v |_2 = 1$ and $#mathrmsupp(v) leq n < m$, where $mathrmsupp$ denotes the support of the vector (i.e. locations of nonzero coordinates). Additionally, assume that $mathrmsupp(u) = mathrmsupp(v)$. Assume that $u^i$ denotes the left-circular shift of $u$ by $i$ locations, i.e.



$$
u = (u_1, dots, u_n, 0, dots, 0) Rightarrow u^1 =
(0, u_2, dots, u_n, 0 dots, u_1)
$$



Form the following matrix:



$$
X = u v^top + u^1 v^1^top + dots + u^m-1 v^m-1^top
$$




Question:
I want to obtain a nontrivial bound on the operator and Frobenius norms of this matrix. The only bound that I was able to obtain so far is using the subadditivity of the Frobenius norm and the fact that $| u_i v_i^top |_F = 1, ; forall i.$
$$
| X |_mathrmop leq | X |_F leq m | u_i v_i^top | = m
$$




Any ideas or pointers to references are more than welcome.




linear-algebra matrices reference-request spheres matrix-norms




share|cite|improve this question











Consider two vectors $u, v in mathbbR^m$ satisfying $| u |_2 = | v |_2 = 1$ and $#mathrmsupp(v) leq n < m$, where $mathrmsupp$ denotes the support of the vector (i.e. locations of nonzero coordinates). Additionally, assume that $mathrmsupp(u) = mathrmsupp(v)$. Assume that $u^i$ denotes the left-circular shift of $u$ by $i$ locations, i.e.



$$
u = (u_1, dots, u_n, 0, dots, 0) Rightarrow u^1 =
(0, u_2, dots, u_n, 0 dots, u_1)
$$



Form the following matrix:



$$
X = u v^top + u^1 v^1^top + dots + u^m-1 v^m-1^top
$$




Question:
I want to obtain a nontrivial bound on the operator and Frobenius norms of this matrix. The only bound that I was able to obtain so far is using the subadditivity of the Frobenius norm and the fact that $| u_i v_i^top |_F = 1, ; forall i.$
$$
| X |_mathrmop leq | X |_F leq m | u_i v_i^top | = m
$$




Any ideas or pointers to references are more than welcome.





linear-algebra matrices reference-request spheres matrix-norms





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 23 at 23:22









VHarisop

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  • 1




    I'd expect the worst case to be constant $u$ and $v$ on a contiguous support. That would yield a bound of roughly $sqrtnm$ for the Frobenius norm.
    – joriki
    Jul 24 at 4:08












  • 1




    I'd expect the worst case to be constant $u$ and $v$ on a contiguous support. That would yield a bound of roughly $sqrtnm$ for the Frobenius norm.
    – joriki
    Jul 24 at 4:08







1




1




I'd expect the worst case to be constant $u$ and $v$ on a contiguous support. That would yield a bound of roughly $sqrtnm$ for the Frobenius norm.
– joriki
Jul 24 at 4:08




I'd expect the worst case to be constant $u$ and $v$ on a contiguous support. That would yield a bound of roughly $sqrtnm$ for the Frobenius norm.
– joriki
Jul 24 at 4:08















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