### Research Experiences for Undergraduates (REU) Seminar

Topic: Polynomial Equations Over Matrices

Speaker: **Robert Wilson**, Associate Dean and Professor of Mathematics, Rutgers University

Date and time: Friday, June 29, 2001, 12:00 noon

Location: DIMACS Center, Room 431, Rutgers University, Piscataway, NJ

Lunch will be served

A well-known result of elementary algebra states that if *a* and *b* are complex numbers, then the quadratic equation

*x*^{2} + *ax* + *b* = 0

has at most two solutions in complex numbers.

We will consider the same problem where *a, b,* and *x* are two-by-two matrices over the complex numbers.
We will solve this problem using basic ideas from linear algebra: eigenvectors, eigenvalues and determinants.
The solution generalizes to equations of degree *m*

*x*^{m} + *a*_{m-1}x^{m-1} + ... + *a*_{1}x + *a*_{0} = 0

where *x* and the *a*_{i} are *n*-by-*n* matrices over the complex numbers.

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