Sum of $k$ smallest singular values
$begingroup$
The $k$th Ky Fan norm $lVertcdotrVert_{(k)}$ is defined as the sum of the $k$ largest singular values. Furthermore, for an $mtimes n$ matrix $A$
$$
lVert ArVert_{(k)} = max_{UU^*=VV^*=I_k}|text{tr}(UAV)|.
$$
For Hermitian matrices the sum of the $k$ smallest eigenvalues, denoted $E_k(H)$, is
$$
E_k(H) = min_{UU^*=I_k}text{tr}(UHU^*),
$$
so I was wondering if the sum of the $k$ smallest singular values $S_k(A)$ for a matrix $A$ can be written
$$
S_k(A) = min_{UU^*=VV^*=I_k}|text{tr}(UAV)|.
$$
As I have not seen this expression I thought that perhaps it is not true, and that the argument used for the two previous examples fails at some point. Could anyone shed some light on this?
matrices matrix-norms
asked Dec 16 '18 at 22:24
PeterAPeterA
989
$endgroup$
add a comment |
$begingroup$
The $k$th Ky Fan norm $lVertcdotrVert_{(k)}$ is defined as the sum of the $k$ largest singular values. Furthermore, for an $mtimes n$ matrix $A$
$$
lVert ArVert_{(k)} = max_{UU^*=VV^*=I_k}|text{tr}(UAV)|.
$$
For Hermitian matrices the sum of the $k$ smallest eigenvalues, denoted $E_k(H)$, is
$$
E_k(H) = min_{UU^*=I_k}text{tr}(UHU^*),
$$
so I was wondering if the sum of the $k$ smallest singular values $S_k(A)$ for a matrix $A$ can be written
$$
S_k(A) = min_{UU^*=VV^*=I_k}|text{tr}(UAV)|.
$$
As I have not seen this expression I thought that perhaps it is not true, and that the argument used for the two previous examples fails at some point. Could anyone shed some light on this?
matrices matrix-norms
asked Dec 16 '18 at 22:24
PeterAPeterA
989
$endgroup$
$begingroup$
Presumably you mean the sum of the $k$ smallest singular values
$endgroup$
– Omnomnomnom
Dec 16 '18 at 22:55
$begingroup$
Thank you, it has been corrected.
$endgroup$
– PeterA
Dec 17 '18 at 8:08
add a comment |
$begingroup$
The $k$th Ky Fan norm $lVertcdotrVert_{(k)}$ is defined as the sum of the $k$ largest singular values. Furthermore, for an $mtimes n$ matrix $A$
$$
lVert ArVert_{(k)} = max_{UU^*=VV^*=I_k}|text{tr}(UAV)|.
$$
For Hermitian matrices the sum of the $k$ smallest eigenvalues, denoted $E_k(H)$, is
$$
E_k(H) = min_{UU^*=I_k}text{tr}(UHU^*),
$$
so I was wondering if the sum of the $k$ smallest singular values $S_k(A)$ for a matrix $A$ can be written
$$
S_k(A) = min_{UU^*=VV^*=I_k}|text{tr}(UAV)|.
$$
As I have not seen this expression I thought that perhaps it is not true, and that the argument used for the two previous examples fails at some point. Could anyone shed some light on this?
matrices matrix-norms
asked Dec 16 '18 at 22:24
PeterAPeterA
989
$endgroup$
The $k$th Ky Fan norm $lVertcdotrVert_{(k)}$ is defined as the sum of the $k$ largest singular values. Furthermore, for an $mtimes n$ matrix $A$
$$
lVert ArVert_{(k)} = max_{UU^*=VV^*=I_k}|text{tr}(UAV)|.
$$
For Hermitian matrices the sum of the $k$ smallest eigenvalues, denoted $E_k(H)$, is
$$
E_k(H) = min_{UU^*=I_k}text{tr}(UHU^*),
$$
so I was wondering if the sum of the $k$ smallest singular values $S_k(A)$ for a matrix $A$ can be written
$$
S_k(A) = min_{UU^*=VV^*=I_k}|text{tr}(UAV)|.
$$
As I have not seen this expression I thought that perhaps it is not true, and that the argument used for the two previous examples fails at some point. Could anyone shed some light on this?
matrices matrix-norms
matrices matrix-norms
asked Dec 16 '18 at 22:24
PeterAPeterA
989
asked Dec 16 '18 at 22:24
PeterAPeterA
989
edited Dec 17 '18 at 8:08
PeterA
asked Dec 16 '18 at 22:24
PeterAPeterA
989
asked Dec 16 '18 at 22:24
PeterAPeterA
989
asked Dec 16 '18 at 22:24
PeterAPeterA
989
989
$begingroup$
Presumably you mean the sum of the $k$ smallest singular values
$endgroup$
– Omnomnomnom
Dec 16 '18 at 22:55
$begingroup$
Thank you, it has been corrected.
$endgroup$
– PeterA
Dec 17 '18 at 8:08
add a comment |
$begingroup$
Presumably you mean the sum of the $k$ smallest singular values
$endgroup$
– Omnomnomnom
Dec 16 '18 at 22:55
$begingroup$
Thank you, it has been corrected.
$endgroup$
– PeterA
Dec 17 '18 at 8:08
$begingroup$
Presumably you mean the sum of the $k$ smallest singular values
$endgroup$
– Omnomnomnom
Dec 16 '18 at 22:55
$begingroup$
Presumably you mean the sum of the $k$ smallest singular values
$endgroup$
– Omnomnomnom
Dec 16 '18 at 22:55
$begingroup$
Thank you, it has been corrected.
$endgroup$
– PeterA
Dec 17 '18 at 8:08
$begingroup$
Thank you, it has been corrected.
$endgroup$
– PeterA
Dec 17 '18 at 8:08
add a comment |
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$begingroup$
Presumably you mean the sum of the $k$ smallest singular values
$endgroup$
– Omnomnomnom
Dec 16 '18 at 22:55
$begingroup$
Thank you, it has been corrected.
$endgroup$
– PeterA
Dec 17 '18 at 8:08