How to determine a rank of quadratic form if on the diagonal there is all zeros?

[Zappa]

As far as I know, in order to find a rank of any matrix you need to get 1's on the diagonal and zero's below the 1's, so how should I proceed if it's all zeros to begin with on the diagonal? The polynomial from where I derive the matrix is as follows, - f = x1x2 - x2x3. ( the numbers are obviously indices.) So there is no squared argument.

 

[tkhunny]

I might think on the Singular Value Decomposition.

[math]M = U\;\cdot\;\Sigma\;\cdot\;V^{T}[/math], Where:

[MATH] M= \left[ {\begin{array}{cc} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & -1 & 0 \\ \end{array} } \right] [/MATH]
[MATH] U= \left[ {\begin{array}{cc} 0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 0 \\ \end{array} } \right] [/MATH]
[MATH] \Sigma= \left[ {\begin{array}{cc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ \end{array} } \right] [/MATH]
[MATH] V= \left[ {\begin{array}{cc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right] [/MATH]
This sort of thing used to be hard. Various software can churn it out in a lot less time than it takes to enter the data.

 

[Zappa]

Thanks for input, but how did you get all those matrices?

 

[tkhunny]

You supplied M.

For the others, please see the last two sentences.

There are also videos of course lectures if you like. Here's one:

 

[Steven G]

As far as I know, in order to find a rank of any matrix you need to get 1's on the diagonal and zero's below the 1's, so how should I proceed if it's all zeros to begin with on the diagonal?

Interchanging two rows (or columns) should help. It is not guaranteed to help, for example what if all entries are 0.

What information does the rank give you?? You must know this answer! Now will interchanging two or more rows affect the rank? How about interchanging columns? What about interchanging rows and columns??