Solving an equation including matrices

[Hokey]

Homework Statement

Okay. So I have an equation:

ABA + BAB = 2I

where A and B are square nxn-matrices and I is the identity matrix. From this, I am supposed to find a way to express B as a function of A (given that A is close to I). So B = F(A), and it is also given that F(I) = I.

Homework Equations

The Attempt at a Solution

I just haven't ever come across an equation where there are matrices involved (apart from Ax = B...), and it doesn't really get me anywhere when I try multiplying with A or B's inverse from left, right or center.

The only thing I can think of is that since A is supposed to be close to I, maybe I can say that A = I + H. When inserting this on the left side of the equation, I get B + B^2 + {a couple of terms with H in them so I ignore them, assuming that they are small}. But I'm not sure where to go from B + B^2 = 2I either. A is gone, so now I can't find a function that depends on A, so it seems like I'm on the wrong track...

Help, please? I just want to know what you can and can't do when it comes to matrices in equations, since I've never seen an example of how you solve things like this! Thanks!

 

[Dickfore]

Your matrix equation is quadratic in $B$ due to the term $B A B$. How would you solve a simpler equation:

$$X^{2} = A$$

where $A$ is a square matrix?

 

[Hokey]

How would you solve a simpler equation:

$$X^{2} = A$$

where $A$ is a square matrix?

Umm... it's a square matrix, does that mean that we can decide the square root? :P

Like I said, I haven't really done anything like this before. If anyone can recommend some online reading (preferrably with examples of calculations) about solving matrix equations it would be much appreciated :) When I google it I get mostly hits on numerical calculations, not a lot about handling these things symbolically.

 

[I like Serena]

Your matrix equation is quadratic in $B$ due to the term $B A B$. How would you solve a simpler equation:

$$X^{2} = A$$

where $A$ is a square matrix?

I spent some thought on this and decided to work out an example.

If we take A to represent a half-turn, its matrix is:

$$A = \begin{pmatrix}-1 & 0 \\ 0 & -1 \end{pmatrix}$$

The equation has an infinite number of solutions. For instance:

$$X = \begin{pmatrix}1 & 1 \\ -2 & -1 \end{pmatrix} \qquard \textrm{ or } \qquad X = \begin{pmatrix}0 & -1 \\ 1 & 0 \end{pmatrix}$$

Could you elaborate?

 

[Starlah]

Hey, I am also very interested in solving this equation.