Proving Lower Triangular Matrices When i > j

[HMPARTICLE]

Homework Statement

Prove that is $A$ is lower triangular and $B_{ij}$ is the matrix that results when the ith row and jth column of A are deleted, then $B_{ij}$ is lower triangular if i > j.

Homework Equations

The Attempt at a Solution

I know that a square matrix is lower triangular if and only if the jth column starts with at least j-1 zero's for every j.

I am attemting to prove this by contradiction.
If i = j, then the jth column of B has j-1 zeros, but this is true?
if i was to take a 4 x 4 lower triangluar matrix and delete its first row and first column would the 1st,2nd,3rd,4th columns have 0,1,2,3 zeros respecively?Lets ignore that, prehaps I am saying/doing something silly, to finish my proof by contradiction i also have to show that for i < j, we don't have a lower triangular matrix.

if the ith row and the (i+n)th collumn, where n is a postitive integer, of A is deleted, then the ...

ah I am really not getting this one guys. I am lost.

 

[HMPARTICLE]

Have i not followed the forums rules? if not could someone please point out the error of my ways

 

[Ray Vickson]

Have i not followed the forums rules? if not could someone please point out the error of my ways

I think you have not done "enough" work on your own to satisfy PF requirements. You ask: "if i was to take a 4 x 4 lower triangluar matrix and delete its first row and first column would the 1st,2nd,3rd,4th columns have 0,1,2,3 zeros respecively?" I do not see why you cannot answer this for yourself: just draw a sketch of what a 4 x 4 lower triangular matrix must look like, then remove its first row and column. What does it look like now? Try to generalize this to removing row $r$ and column $c$ from a lower-triangular matrix, where $r = c$. What does the new matrix look like? Ditto if $r > c$.

 

[HMPARTICLE]

Thanks, Ray. Maybe i should have made what i HAD done already a little more clear.

I have already done what you said, regarding a 4x4 matrix and eliminating the first row and column, what results is another lower triangular matrix. the question asks me to show that that if i > j then $B_{ij}$ is a lower triangluar matrix. but in the 4 x 4 example, when i = j, $B_{ij}$ is a lower triangular still, so is proof by contradtiction not the way to go?

in summary;
I have found by experiment, when i = j, $B_{ij}$ is still lower triangular
Also using a 4 x4 matrix as an experiment, when the first row and second column are deleted, what results is another lower triangular matrix. that is i < j

finally, when the second row and the first column are deleted, that is i > j, now, this is not lower triangular, since the first row is a row of zeros.

Prehaps I am not understanding the question, but what is evedent from experiment is that the opposite of what i have to prove is true.
I hope my definition of a lower triangular is correct;
that is, each entry to the right of the main diagonal is zero.

That is where my confusion lies.

 

[Ray Vickson]

Thanks, Ray. Maybe i should have made what i HAD done already a little more clear.

I have already done what you said, regarding a 4x4 matrix and eliminating the first row and column, what results is another lower triangular matrix. the question asks me to show that that if i > j then $B_{ij}$ is a lower triangluar matrix. but in the 4 x 4 example, when i = j, $B_{ij}$ is a lower triangular still, so is proof by contradtiction not the way to go?

in summary;
I have found by experiment, when i = j, $B_{ij}$ is still lower triangular
Also using a 4 x4 matrix as an experiment, when the first row and second column are deleted, what results is another lower triangular matrix. that is i < j

finally, when the second row and the first column are deleted, that is i > j, now, this is not lower triangular, since the first row is a row of zeros.

Prehaps I am not understanding the question, but what is evedent from experiment is that the opposite of what i have to prove is true.
I hope my definition of a lower triangular is correct;
that is, each entry to the right of the main diagonal is zero.

That is where my confusion lies.

When you omit row 2 and column 1 you get a matrix that is again lower triangular. The fact that the first row is all zero is irrelevent; all that matters is that elements strictly to the right of the diagonal are zero---and they are in your case.