Solving for constant in a linear combination of vectors

[robbondo]

Homework Statement

P = c1*V1 + c2*V2
Where P, V1, and V2, are equal sized matrices

Homework Equations

The Attempt at a Solution

So what this problem amounts to is me trying to find the steady state of Markov matrix. So I solved for the eigen vectors, and as is my understanding I should be able to solve for my initial condition matrix P as a linear combination of some constant and the eigen vectors. I'm doing this all in MATLAB and I can't seem to figure out how to plug that equation into MATLAB. I know that it should be possible to solve for these constants(C1,C2) by hand, but I'd like to be able to do this for large matrices. Any suggestions?

 

[dirk_mec1]

Is this matrix P a probability-matrix?

 

[robbondo]

yeah... So I figured out that i can make the vectors and the constants into matrices and then use the inverse to solve for C1, etc. but it's not working for some reason. I'm still working on it.

 

[Acetv]

One way you can think about it is that your matrices $$P$$, $$V_1$$ and $$V_2$$ are all vectors in $$\mathbb{R}^{n}$$.

So we have the equation,

$$P = c_1 V_1 + c_2 V_2$$

Taking the inner product of $$P$$ with the $$V_i$$ you get a system of equations for the $$c_{i}$$.

$$\langle P,V_1 \rangle = c_1 \langle V_1,V_1 \rangle + c_2 \langle V_2,V_1 \rangle$$
$$\langle P,V_2 \rangle = c_1 \langle V_1,V_2 \rangle + c_2 \langle V_2,V_2 \rangle$$

Further, if your $$V_i$$ are orthonormal, then the above reduces to,

$$\langle P,V_i \rangle = c_i$$

You should be able to solve the above system ($$A x=b$$) by left-multiplying both sides by $$A^{-1}$$ since the only time $$A$$ would be non-invertible is if $$\langle V_1,V_1 \rangle \langle V_2,V_2 \rangle = {\langle V_1,V_2 \rangle}^2$$;that is, if $$V_1$$ and $$V_2$$ are linearly dependent. However, this is impossible because they are distinct eigenvectors of a matrix.