Given a symmetric tridiagonal matrix in the form of

$$\left( {\begin{array}{*{20}{c}} {{x_1}}&{{y_1}}&{}&{} \\ {{y_1}}&{{x_2}}& \ddots &{} \\ {}& \ddots & \ddots &{{y_{n - 1}}} \\ {}&{}&{{y_{n - 1}}}&{{x_n}} \end{array}} \right)$$

where $y_1,...,y_{n-1}$ are all positive numbers. My questions is, does the maximum eigenvalue of its leading principal submatrices $\left( {{x_1}} \right)$, $\left( {\begin{array}{*{20}{c}} {{x_1}}&{{y_1}} \\ {{y_1}}&{{x_2}} \end{array}} \right)$, $\left( {\begin{array}{*{20}{c}} {{x_1}}&{{y_1}}&{} \\ {{y_1}}&{{x_2}}&{{y_2}} \\ {}&{{y_2}}&{{x_3}} \end{array}} \right)$, etc. are non-decreasing? If so, which theorem states this or can be used to prove this? Thanks!!


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