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How can one simply see that Ky Fan $k$-norm satisfies the triangle inequality? (The Ky Fan $k$-norm of a matrix is the sum of the $k$ largest singular values of the matrix)


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Could you please define the Ky Fan $k$-norm and let us know what you already tried? Also, is this homework? – Glen Wheeler Mar 21 '11 at 19:17
The Ky Fan $k$-norm of a matrix is the sum of the $k$ largest singular values of the matrix – user4727 Mar 21 '11 at 19:28
@user4727 Thanks for the edit. Are you aware of the original papers by Ky Fan (~1950)? The reason why it is called the "Ky Fan norm" is because he proved it is a norm first. This might be a good opportunity for some "research"! :) – Glen Wheeler Mar 21 '11 at 19:47

For any $k$-plane $U$ in $\mathbb{C}^n$, let $i_U$ be the inclusion of $U$ into $\mathbb{C}^n$ and let $p_U$ be the orthogonal projection of $\mathbb{C}^n$ onto $U$.

Lemma: Let the singular values of $A$ be $\sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_n$. Then $\max_{U, V} \ \mathrm{Tr} \left( p_V \circ A \circ i_U \right)= \sigma_1+ \cdots +\sigma_k$, where the max is over all pairs of $k$-planes in $\mathbb{C}^n$.

Proof is left for the reader, using his or her favorite definition of singular values.

Then $\max_{U, V} \ \mathrm{Tr} \left(p_V \circ (A+B) \circ i_U\right) \leq \max_{U, V} \ \mathrm{Tr} \left( p_V \circ A \circ i_U \right) + \max_{U, V} \ \mathrm{Tr} \left( p_V \circ B \circ i_U \right)$.

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