ABSTRACT
For the Wigner matrix and the group of circulant matrices, the number of times each random variable appears in the matrix is the same across most variables but for asymptotically negligible exceptions. We may call them balanced matrices. For them, the LSDs exist after the eigenvalues are scaled by n - 1 / 2 $ n^{-1/2} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429488436/725f3a92-cccc-4b3b-961a-84021a8d495e/content/inline-math6_1.tif"/> . However, the Toeplitz and Hankel matrices are unbalanced. It seems natural to consider their balanced versions where each entry is scaled by a constant multiple of the square root of the number of times that entry appears in the matrix instead of the uniform scaling by n - 1 / 2 $ n^{-1/2} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429488436/725f3a92-cccc-4b3b-961a-84021a8d495e/content/inline-math6_2.tif"/> . Define the (symmetric) balanced Hankel and Toeplitz matrices B H n $ BH_{n} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429488436/725f3a92-cccc-4b3b-961a-84021a8d495e/content/inline-math6_3.tif"/> and B T n $ BT_{n} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429488436/725f3a92-cccc-4b3b-961a-84021a8d495e/content/inline-math6_4.tif"/> with input { x i } $ \{x_i\} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429488436/725f3a92-cccc-4b3b-961a-84021a8d495e/content/inline-math6_5.tif"/> as follows: B H n = x 1 1 x 2 2 x 3 3 … x n - 1 n - 1 x n n x 2 2 x 3 3 x 4 4 … x n n x n + 1 n - 1 x 3 3 x 4 4 x 5 5 … x n + 1 n - 1 x n + 2 n - 2 ⋮ x n n x n + 1 n - 1 x n + 2 n - 2 … x 2 n - 2 2 x 2 n - 1 1 . $$ \begin{aligned} BH_{n}\ = \left[ \begin{array} {cccccc} \frac{x_{1}}{\sqrt{1}}&\frac{x_{2}}{\sqrt{2}}&\frac{x_{3}}{\sqrt{3}}&\ldots&\frac{x_{n-1}}{\sqrt{n-1}}&\frac{x_{n}}{\sqrt{n}} \\ \frac{x_{2}}{\sqrt{2}}&\frac{x_{3}}{\sqrt{3}}&\frac{x_{4}}{\sqrt{4}}&\ldots&\frac{x_{n}}{\sqrt{n}}&\frac{x_{n+1}}{\sqrt{n-1}} \\ \frac{x_{3}}{\sqrt{3}}&\frac{x_{4}}{\sqrt{4}}&\frac{x_{5}}{\sqrt{5}}&\ldots&\frac{x_{n+1}}{\sqrt{n-1}}&\frac{x_{n+2}}{\sqrt{n-2}} \\&\,&\vdots&\,&\\ \frac{x_{n}}{\sqrt{n}}&\frac{x_{n+1}}{\sqrt{n-1}}&\frac{x_{n+2}}{\sqrt{n-2}}&\ldots&\frac{x_{2n-2}}{\sqrt{2}}&\frac{x_{2n-1}}{\sqrt{1}} \end{array} \right]. \end{aligned} $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429488436/725f3a92-cccc-4b3b-961a-84021a8d495e/content/math6_1.tif"/> B T n = x 0 n x 1 n - 1 x 2 n - 2 … x n - 2 2 x n - 1 1 x 1 n - 1 x 0 n x 1 n - 1 … x n - 3 3 x n - 2 2 x 2 n - 2 x 1 n - 1 x 0 n … x n - 4 4 x n - 3 3 ⋮ x n - 1 1 x n - 2 2 x n - 3 3 … x 1 n - 1 x 0 n . $$ \begin{aligned} BT_{n}\ = \left[ \begin{array} {cccccc} \frac{x_{0}}{\sqrt{n}}&\frac{x_{1}}{\sqrt{n -1}}&\frac{x_{2}}{\sqrt{n - 2}}&\ldots&\frac{x_{n-2}}{\sqrt{2}}&\frac{x_{n-1}}{\sqrt{1}} \\ \frac{x_{1}}{\sqrt{n -1}}&\frac{x_{0}}{\sqrt{n}}&\frac{x_{1}}{\sqrt{n-1}}&\ldots&\frac{x_{n-3}}{\sqrt{3}}&\frac{x_{n-2}}{\sqrt{2}} \\ \frac{x_{2}}{\sqrt{n-2}}&\frac{x_{1}}{\sqrt{n-1}}&\frac{x_{0}}{\sqrt{n}}&\ldots&\frac{x_{n-4}}{\sqrt{4}}&\frac{x_{n-3}}{\sqrt{3}} \\&\,&\vdots&\,&\\ \frac{x_{n-1}}{\sqrt{1}}&\frac{x_{n-2}}{\sqrt{2}}&\frac{x_{n-3}}{\sqrt{3}}&\ldots&\frac{x_{1}}{\sqrt{n-1}}&\frac{x_{0}}{\sqrt{n}} \end{array} \right]. \end{aligned} $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429488436/725f3a92-cccc-4b3b-961a-84021a8d495e/content/math6_2.tif"/>
