1

I'm not a math major and I find the statement below confusing (from a paper by Candes and Recht on Matrix Completion). Can someone clarify this? I'm sure it's painfully simple/obvious.

"For instance, consider the rank-2 symmetric matrix M given by

enter image description here

where the singular values are arbitrary. Then this matrix vanishes everywhere except in the top left 2x2 corner."

I'm taking e1 and e2 as [1 0] and [0 1], respectively. It's confusing to me because it seems that M was never any bigger than 2x2. So why would they say it vanishes everywhere except in the 2x2 left-uppermost corner?

Rodrigo de Azevedo
  • 23,788
val
  • 1,087
  • 1
    It might be helpful to know the context for these comments. – Igor Rivin Jan 03 '14 at 16:57
  • ok - sure. Bottom of 5th page at http://statweb.stanford.edu/~candes/papers/MatrixCompletion.pdf – val Jan 03 '14 at 16:58
  • 1
    Perhaps $e_1=(1,0,\dots,0)$ and $e_2=(0,1,0,\dots,0)$ are $n$-vectors for arbitrary $n\ge2$. – Gerry Myerson Jan 03 '14 at 17:00
  • What they are saying is that if you know a sample of the entries of the matrix, and the singular values, it is impossible to reconstruct this matrix. – Igor Rivin Jan 03 '14 at 17:00
  • 1
    @GerryMyerson correct, the matrix is $n\times n.$ – Igor Rivin Jan 03 '14 at 17:01
  • OK - thank you both. I guess I had expected the sum to go to n (and not 2). Since the basis vectors would take care of limiting the values to the top left 2x2 corner.... putting the sum up to 2 makes M strictly be 2x2 when what they want you to see is that M is nxn but only the 2x2 upper left corner counts. – val Jan 03 '14 at 17:06
  • no - i think i see an error in my wording in my last comment...it wouldn't work either to put the sum up to n because then you would be looking for u_n. Anyway, THanks again – val Jan 03 '14 at 17:08

1 Answers1

2

You want $$ e_1 = [1, 0, 0, 0, \cdots 0] \\ e_2 = [0,1,0,0, \cdots 0] $$

If each vector is $n$ long you will end up with a $n\times n$ matrix with just two entries as stated.

user44197
  • 9,858