In mathematics, a matrix polynomial is a polynomial with matrices as variables. Examples include:
- where P is a polynomial,
- and I is the identity matrix.
- the commutator of A and B.
A matrix polynomial equation is an equality between two matrix polynomials, which holds for the specific matrices in question. If , (where A is a matrix over a field), then the eigenvalues of A satisfy the characteristic equation[disputed ] .
A matrix polynomial identity is a matrix polynomial equation which holds for all matricies A in a specified matrix ring Mn(R).
Matrix geometrical series
Matrix polynomials can be used to sum a matrix geometrical series as one would an ordinary geometric series,
If I-A is nonsingular one can evaluate the expression for the sum S.
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