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It says in Rao that it's "easy to prove" that a matrix H in hermite canonical form is necessarily idempotent under matrix multiplication (i.e. HxH=H).

I am trying to find a proof to this but everyone claims it's obvious.

It is not so obvious to me!

For example the following matrix H is hermite canonical

H = 1 2
    0 1

But multiplying that by itself yields H x H =

1 2  x  1 2 = 1 4 != H
0 1     0 1   0 1

What am I missing?

Victor Grazi
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    Are you aware of definition A.2.8.2 in this book? According to this definition, a square matrix $H$ is Hermite canonical, if every diagonal entry is either 0 or 1, and if a diagonal entry is 0, then every other entry in that same row is 0 as well, ... – jflipp Feb 01 '15 at 15:06
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    [continued] ... and if a diagonal entry is 1, then every other entry in that same column is 0. According to this definition, your example matrix is not Hermite canonical. – jflipp Feb 01 '15 at 15:07
  • I don't follow. None of my diagonal entries are 0. According to the definition in Rao, a 1 in the diagonal means everything below it must be 0. Mine conforms to that – Victor Grazi Feb 01 '15 at 16:09
  • Maybe you didn't read the complete definition of Hermite canonical in Rao. It starts at the bottom of page 18 (see here) and goes on to page 19 (that's important! :-). In particular, it says that a 1 on the diagonal enforces all other elements in that same column to be 0. This last condition doesn't hold for your example. – jflipp Feb 01 '15 at 16:30
  • I think it means all "other" entries are zero, meaning the ones below. According to your definition, the matrix can only have zeros and ones. Or am I wrong? – Victor Grazi Feb 01 '15 at 18:35

1 Answers1

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Apparently my confusion was in assuming that if a diagonal contains a 0, then both its row and column must contain only 0's.

This is not the case. Here are the rules for creating a Hermite canonical matrix

  • Diagonal must consist of only ones and zeros.
  • Everything below the diagonal must be zeros.
  • If a diagonal element is 0, then every other element in that row must also be 0. (Note that in this case, any value may be above the diagonal 0 in the column.)

  • If a diagonal element is 1, then every other entry in that column (above and below) must be 0.

    Here is an example of a matrix in Hermite canonical form:

    1 0 2 0
    0 1 3 0
    0 0 0 0
    0 0 0 1

And multiplying this by itself indeed yields the same matrix.

And it is indeed easy to verify that H*H=H, because the 1 diagonal elements obviously produce the same row (as with the identity matrix I) and the 0 diagonal elements will produce a zero row in the product, since the row in that position is the sum of the row vectors with each row multiplied by the value in the row, i.e, all zeros.

Victor Grazi
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