The Structure of a Quotient Ring modulo x^p-1

Let f(X)=X^p-1 \in F_l[X] for a prime p. We are interested in the structure of the ring R=F_l[X]/(f). Via the Chinese Remainder Theorem, this question is equivalent to finding a factorization of f over F_l. Suppose f factors in F_l as

\displaystyle f(X) = (X-1) \prod_i^s{q_i(X)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \   (*)

where each q_i is an irreducible polynomial. What are the degrees of q_1, \cdots, q_s? What is s?

We claim that R=F_l \oplus (F_{l^r})^s, where r is the multiplicative order of l mod p.

This structure has been used in a beautifully instructive paper by Evra, Kowalski, and Lubotzky. (Yes, the one in Lubotzky-Philip-Sarnak.) In this paper, they have established connections among the Fourier transform, uncertainty principle, and dimension of ideals generated by polynomials in the ring R above. We’ll talk about these connections in another post. The content of this post is Proposition 2.6(3) from their paper.

Continue reading “The Structure of a Quotient Ring modulo x^p-1”

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In a Hermitian Matrix, the Eigenvectors of Different Eigenvalues are Orthogonal

This is an elementary (yet important) fact in matrix analysis.

Statement

Let M be an n\times n complex Hermitian matrix which means M=M^* where * denotes the conjugate transpose operation. Let \lambda_1, \neq \lambda_2 be two different  eigenvalues of M. Let x, y be the two eigenvectors of M corresponding to the two eigenvalues \lambda_1 and \lambda_2, respectively.

Then the following is true:

\boxed{\lambda_1 \neq \lambda_2 \iff \langle x, y \rangle = 0.}

Here \langle a,b\rangle denotes the usual inner product of two vectors x,y, i.e.,

\langle x,y \rangle := y^*x.

Continue reading “In a Hermitian Matrix, the Eigenvectors of Different Eigenvalues are Orthogonal”