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Sum of $k$ smallest singular values

3

$begingroup$

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 $A$
$$
lVert ArVert_(k) = max_UU^*=VV^*=I_k|texttr(UAV)|.
$$
For Hermitian matrices the sum of the $k$ smallest eigenvalues, denoted $E_k(H)$, is
$$
E_k(H) = min_UU^*=I_ktexttr(UHU^*),
$$
so I was wondering if the sum of the $k$ smallest singular values $S_k(A)$ for a matrix $A$ can be written
$$
S_k(A) = min_UU^*=VV^*=I_k|texttr(UAV)|.
$$
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 singularvalues matrix-norms

share|cite|improve this question

edited Mar 31 at 7:37

Rodrigo de Azevedo

13.2k41962

asked Dec 16 '18 at 22:24

PeterAPeterA

1039

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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 | 

3

$begingroup$

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 $A$
$$
lVert ArVert_(k) = max_UU^*=VV^*=I_k|texttr(UAV)|.
$$
For Hermitian matrices the sum of the $k$ smallest eigenvalues, denoted $E_k(H)$, is
$$
E_k(H) = min_UU^*=I_ktexttr(UHU^*),
$$
so I was wondering if the sum of the $k$ smallest singular values $S_k(A)$ for a matrix $A$ can be written
$$
S_k(A) = min_UU^*=VV^*=I_k|texttr(UAV)|.
$$
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 singularvalues matrix-norms

share|cite|improve this question

edited Mar 31 at 7:37

Rodrigo de Azevedo

13.2k41962

asked Dec 16 '18 at 22:24

PeterAPeterA

1039

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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 $A$
$$
lVert ArVert_(k) = max_UU^*=VV^*=I_k|texttr(UAV)|.
$$
For Hermitian matrices the sum of the $k$ smallest eigenvalues, denoted $E_k(H)$, is
$$
E_k(H) = min_UU^*=I_ktexttr(UHU^*),
$$
so I was wondering if the sum of the $k$ smallest singular values $S_k(A)$ for a matrix $A$ can be written
$$
S_k(A) = min_UU^*=VV^*=I_k|texttr(UAV)|.
$$
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 singularvalues matrix-norms

share|cite|improve this question

edited Mar 31 at 7:37

Rodrigo de Azevedo

13.2k41962

asked Dec 16 '18 at 22:24

PeterAPeterA

1039

$endgroup$

share|cite|improve this question

edited Mar 31 at 7:37

Rodrigo de Azevedo

13.2k41962

asked Dec 16 '18 at 22:24

PeterAPeterA

1039

share|cite|improve this question

edited Mar 31 at 7:37

Rodrigo de Azevedo

13.2k41962

asked Dec 16 '18 at 22:24

PeterAPeterA

1039

edited Mar 31 at 7:37

Rodrigo de Azevedo

13.2k41962

edited Mar 31 at 7:37

Rodrigo de Azevedo

13.2k41962

asked Dec 16 '18 at 22:24

PeterAPeterA

1039

asked Dec 16 '18 at 22:24

PeterAPeterA

1039