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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2005 • Question 3

Question

Math 6310 • Algebra Spring 2005 • Question 3

Let \(A\) be an \(n\)-by-\(n\) diagonal matrix with coefficients in a field \(F\). Suppose the characteristic polynomial of \(A\) has the form \[ (x-c_1)^{d_1}(x-c_2)^{d_2}\cdots (x-c_k)^{d_k} \] with \(c_1,\ldots ,c_k\) distinct. Let \(V\) denote the space of all \(n\)-by-\(n\) matrices over \(F\) which commute with \(A\). Compute the dimension of \(V\) as a vector space over \(F\).

Raw LaTeX

Let \(A\) be an \(n\)-by-\(n\) diagonal matrix with coefficients in a field \(F\). Suppose the
characteristic polynomial of \(A\) has the form
\[
(x-c_1)^{d_1}(x-c_2)^{d_2}\cdots(x-c_k)^{d_k}
\]
with \(c_1,\ldots,c_k\) distinct. Let \(V\) denote the space of all \(n\)-by-\(n\) matrices over \(F\)
which commute with \(A\). Compute the dimension of \(V\) as a vector space over \(F\).

Topic Tags

Galois Groups of Polynomials Irreducible Polynomials Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization Polynomial Rings and Ring Homomorphism Kernels