Properties of Frobenius matrix and decomposability

If $D$ is Frobenius matrix, is there any relationship of the matrix being Frobenius matrix with $D$ being $D = (A-I)B^{-1}$ where $A$ is non-negative square matrix?

Also, how is Frobenius matrix used?

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You can read a lot about frobenius matrix in wikipedia en.wikipedia.org/wiki/Frobenius_matrix –  dineshdileep Dec 17 '12 at 11:14