Suppose that $a_1,a_2,\ldots,a_n$ are $n$ distinct real numbers; is the following statement true?
There is a permutation of $a_1,a_2,\ldots,a_n$, namely $b_1,b_2,\ldots,b_n$, such that the determinant of the following matrix is nonzero:
$$
\begin{bmatrix}
b_1&b_2&\cdots&b_n\\
b_2&b_3&\cdots&b_1\\
\vdots&\vdots&\ddots&\vdots\\
b_n&b_1&\cdots&b_{n-1}\\
\end{bmatrix}
$$
(Such a matrix is called a circulant matrix.)
Answer
This statement is not true, without supplementary conditions on the $a_i$'s. Indeed,
suppose the $\sum_{k=1}^na_k=0 $, whatever your permutation is the vector $[1,1,\ldots,1]^T$ is in the kernel of the circulant matrix of the $b_i$'s, and consequently, its determinant is $0$.
No comments:
Post a Comment