Solving a system of matrix equations numerically

I have four matrix equations including one unknown column vector w that I want to solve for. I know how to solve a system in of linear equations in matrix form. However, in this case I have different equations in matrix form and I am not sure how to solve this especially since there is an integral over a matrix exponential in two of the equations. The system looks as follows (this is not the actual code but rather the problem as it is written down):

A*w = B*w
C*w = D*w
m*integral(expm(Lambda*x),0,N)*w = 0
(E + F*integral(expm(Lambda*x),0,N)) = 1

A, B, C, D, E, and F are known matrices, m is a row vector and N a parameter. I do not expect any complete solutions. I just need a hint how to solve the whole system (preferably without symbolic toolbox) and how to cope with the matrix exponential since I am new to Matlab. Thanks in advance!

[Torsten]

x is a scalar, Lambda is a dense matrix and integration of the resulting matrix is elementwise ?

Or maybe Lambda is a diagonal matrix ?

And the last equation holds of every element of the matrix (E + F*integral(expm(Lambda*x),0,N)) ?

Or "1" is the identity matrix ?

So many questions ...

Best wishes

Torsten.

[Christoph Müller]

Yes, x is a scalar and the integration of the resulting matrix is elementwise.

Lambda is not a diagonal matrix but a dense matrix.

"1" is defintely a scalar. The last equation is a normalization equation (written in matrix form), i.e., all probabilities must sum up to 1. I tried to simplify the original notation with the intention not to cause confusion but I might have done a bad job there. I attached the original equation hoping that this is more clear.

Thanks and best wishes!

Christoph