- #1
Constantinos
- 83
- 1
Hey!
I found this interesting theorem in a textbook, but I was unable to find a proof for it neither in the web nor on my own
The rank of a block triangular matrix is at least and can be greater than the triangular blocks. proof?
specificaly, look here: pp. 25
http://books.google.gr/books?id=PlY...ce=gbs_ge_summary_r&cad=0#v=onepage&q&f=false
[itex] M_n [/itex] is the field [itex]R^{n \times n} [/itex] in the book.
Well I proved the equality claim if the matrix is block diagonal. The proof should go like this:
Take a maximal set of linear independent rows from each diagonal block and combine them. The resulting set is a set of linear independent rows for the whole matrix. Moreover it is maximal since if it could grow larger, then some of the sets we chose for the diagonal blocks would not be maximal. (I can get more precise if need be)
For the block triangular matrix though, I'm pretty clueless... I tried a similar strategy as the above, but didn't work, tried proving by contradiction but again failed (and I don't think it can happen, the resulting inequalities didn't lead me anywhere) and also tried using the subadditivity property of ranks but this also got me nowhere.
Can anyone help me with that? No homework question, independent research.
Thanks!
I found this interesting theorem in a textbook, but I was unable to find a proof for it neither in the web nor on my own
Homework Statement
The rank of a block triangular matrix is at least and can be greater than the triangular blocks. proof?
specificaly, look here: pp. 25
http://books.google.gr/books?id=PlY...ce=gbs_ge_summary_r&cad=0#v=onepage&q&f=false
Homework Equations
[itex] M_n [/itex] is the field [itex]R^{n \times n} [/itex] in the book.
The Attempt at a Solution
Well I proved the equality claim if the matrix is block diagonal. The proof should go like this:
Take a maximal set of linear independent rows from each diagonal block and combine them. The resulting set is a set of linear independent rows for the whole matrix. Moreover it is maximal since if it could grow larger, then some of the sets we chose for the diagonal blocks would not be maximal. (I can get more precise if need be)
For the block triangular matrix though, I'm pretty clueless... I tried a similar strategy as the above, but didn't work, tried proving by contradiction but again failed (and I don't think it can happen, the resulting inequalities didn't lead me anywhere) and also tried using the subadditivity property of ranks but this also got me nowhere.
Can anyone help me with that? No homework question, independent research.
Thanks!