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Not trained in math the solution to this problem is not immediately apparent, plus I am working on a larger problem which I'd rather get to.

I'm trying to isolate the middle value from a $3\times 3$ matrix

Suppose my matrix is

$$\begin{pmatrix} \\ 5 & 7 & 3 \\ 4 & 13 & 9 \\ 9 & 9 & 1 \end{pmatrix}$$

I'd like to produce a matrix using linear algebra methods to get

$$\begin{pmatrix} \\ 0 & 0 & 0 \\ 0 & 13 & 0 \\ 0 & 0 & 0 \end{pmatrix}$$

I do not want a simple answer like subtract

$$\begin{pmatrix} \\ 5 & 7 & 3 \\ 4 & 0 & 9 \\ 9 & 9 & 1 \end{pmatrix}$$

Additionally, what is the terminology for this kind of derived matrix if there is one?

$$\begin{pmatrix} \\ 0 & 0 & 0 \\ 0 & 13 & 0 \\ 0 & 0 & 0 \end{pmatrix}$$

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If subtraction isn't allowed, which operations would you allow? –  Stefan Hansen Jan 16 '13 at 20:01
    
Is multiplication by other matrices allowed? –  Git Gud Jan 16 '13 at 20:01
    
I think it is going to be very hard anyone can help you unless you specify exactly what operations you're allowing on your matrix to "isolate", as you call it, that value. It's clear it cannot be the usual linear operations (substracting/summing one multiple of a row/column to other row/column and etc.), as the given matrix is not similar to the one you want to get, so... –  DonAntonio Jan 16 '13 at 20:02
    
@StefanHansen I don't mind substraction, but it would have to be substraction with a constant matrix. In other words a matrix that can be reused for various 3x3 matrices. –  annoying_squid Jan 16 '13 at 20:02
    
@Git Yes. Anything is allowed accept subtracting the above matrix that I specified. –  annoying_squid Jan 16 '13 at 20:03
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1 Answer 1

up vote 6 down vote accepted

$$\begin{pmatrix}0&0&0\\0&1&0\\0&0&0\end{pmatrix}\begin{pmatrix}5&7&3\\4&13&9\\9&9&1\end{pmatrix}\begin{pmatrix}0&0&0\\0&1&0\\0&0&0\end{pmatrix} = \begin{pmatrix}0&0&0\\0&13&0\\0&0&0\end{pmatrix} $$

The first matrix isolates the middle row and the last isolates the middle column.

Edit: These types of matrices are sometimes called $E_{ij}$, where $i$ is the row and $j$ is the column. So these would be $E_{22}$.

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Bingo. Thanks John. Originally I tried doing it the other way, which is A*B*B, where yours is B*A*B. –  annoying_squid Jan 16 '13 at 20:06
    
No problem. One more thing, if this is the answer you were looking for, could you mark it as accepted? Thanks. –  John Moeller Jan 16 '13 at 20:09
    
BTW, is there a special name for the 3X3 matrix where there is only a 1 in the middle value (row/column)? –  annoying_squid Jan 16 '13 at 20:09
1  
Sometimes these are called $E_{ij}$, where $i$ is the row and $j$ is the column. –  John Moeller Jan 16 '13 at 20:09
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