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Operator identities and the solution of linear matrix difference and differential equations. (English) Zbl 0810.47008

Let \(w(t)\) be a monic polynomial of degree \(n+1\) with complex coefficients, and define the difference quotient \[ w[t,x]:= {w(t)- w(x)\over t- x}. \] This is a symmetric polynomial in \(x\) and \(t\). In this interesting paper, the author uses \(w[t,x]\) with the variables replaced by operators/matrices to obtain identities for powers of matrices and for matrix exponentials. Furthermore, linear matrix difference and differential equations are solved using this idea, and the relationship between matrix equations and polynomial interpolation is explored.

MSC:

47A50 Equations and inequalities involving linear operators, with vector unknowns
47B39 Linear difference operators
39B42 Matrix and operator functional equations
15A24 Matrix equations and identities
39A10 Additive difference equations
34K30 Functional-differential equations in abstract spaces
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