Writing a matrix as sum of a constant * matrix

[2slowtogofast]

Ok I need some help on how I can approach a problem like this.

Say that A B and C are matircies and I know the values of each of them.

And let x and y be constants (ie: 7 or 2 or somthing like that)

What I want to know is can you write

A = xB + yC

How would you check to see if there are an x and y that can make that equation true and if there is how can you find it. I think if B and C are in the span of A then yes there is an x and y but that doest tell me how to find them.

 

[Luka]

You can check if there are $x$ and $y$ that make the equation true by solving the system of equations generated by matrices.

Let matrix A be $A=\left| \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right|$, and B $B=\left| \begin{array}{cc} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array} \right|$. Matrix C will be $C=\left| \begin{array}{cc} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array} \right|$.

You get the system of equations to solve for $x$ and $y$ with the following expression:

$\left| \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right|$ $-$ $x \left| \begin{array}{cc} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array} \right|$ $-$ $y \left| \begin{array}{cc} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array} \right|$ $=$ $\left| \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right|$.

 

[2slowtogofast]

I was thinking of that as well the only problem is what if A B and C are 10 x 10 it would work but it is a lot of work. Is there a more effcient method.

 

[Luka]

Well, you need just a few equations to solve the system, so it doesn't matter if a matrix is 10 x 10. It wouldn't take too long. As far as I know, there is no more efficient method. Today, if we want to deal with 100 x 100 matrices, we use computers. That's why we've made them in the first place. :-)

 

[2slowtogofast]

in your first post how would you expand that to a system of 2 eqns and 2 unkwns. The 2x2 constant matrix is what is screwing me up

 

[HallsofIvy]

You don't expand an equation with two by two matrices into a system of two equations. An equation involving two by two matrices is equivalent to a system of four equations, one for each of the four entries in the matrices. In general, you cannot solve an equation like "A= xB+ yC" for number x and y given matrices A, B, and C unless A, B, and C are carefully chosen, just as you cannot solve four equations in two unknowns unless the equations are "dependent". In general, there simply do NOT exist such numbers.

 

[Luka]

Isn't the solution of the system we get an indicator of the existence of those numbers? If the system has a solution, such $x$ and $y$ exist. Otherwise, they don't.