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Giuseppe Negro
Giuseppe Negro
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I am interested in the following problem: I have an infinite symmetric tridiagonal matrix $$ A_{\infty}= \begin{bmatrix} a_1 & b_1 & & & \\ b_1 & a_2 & b_2 & & \\ & b_2& a_3 & b_3 & \\ & & \ddots & \ddots & \ddots & \\ \end{bmatrix} $$$$ A= \begin{bmatrix} a_1 & b_1 & & & \\ b_1 & a_2 & b_2 & & \\ & b_2& a_3 & b_3 & \\ & & \ddots & \ddots & \ddots & \\ \end{bmatrix} $$ where $a_j, b_j>0$, and I need to determine whether $A_\infty$$A$ is positive definite, meaning that there exists a constantthe corresponding quadratic form is bounded below: $$ Q_A(\beta_1, \beta_2\ldots \beta_n\ldots)\stackrel{\mathrm{def}}{=}\sum_{j=1}^\infty a_j \beta_j^2 + 2b_j\beta_{j}\beta_{j+1} \ge c\sum_{j=1}^\infty \beta_j^2.$$ Here $c_\infty>0$ such$c>0$. (If $c=0$, we say that $$ \sum_{j=1}^\infty a_j \beta_j^2 + 2b_j\beta_{j}\beta_{j+1} \ge c_{\infty}\sum_{j=1}^\infty \beta_j^2$$ $A$ is positive semidefinite).

Question Are there infinite-dimensional versions of the familiar criterions of linear algebra, such as the Sylvester's criterion or the diagonal dominance sufficient condition?

Any result or reference is gladly welcome.

I am interested in the following problem: I have an infinite symmetric tridiagonal matrix $$ A_{\infty}= \begin{bmatrix} a_1 & b_1 & & & \\ b_1 & a_2 & b_2 & & \\ & b_2& a_3 & b_3 & \\ & & \ddots & \ddots & \ddots & \\ \end{bmatrix} $$ where $a_j, b_j>0$, and I need to determine whether $A_\infty$ is positive definite, meaning that there exists a constant $c_\infty>0$ such that $$ \sum_{j=1}^\infty a_j \beta_j^2 + 2b_j\beta_{j}\beta_{j+1} \ge c_{\infty}\sum_{j=1}^\infty \beta_j^2$$

Question Are there infinite-dimensional versions of the familiar criterions of linear algebra, such as the Sylvester's criterion or the diagonal dominance sufficient condition?

Any result or reference is gladly welcome.

I am interested in the following problem: I have an infinite symmetric tridiagonal matrix $$ A= \begin{bmatrix} a_1 & b_1 & & & \\ b_1 & a_2 & b_2 & & \\ & b_2& a_3 & b_3 & \\ & & \ddots & \ddots & \ddots & \\ \end{bmatrix} $$ where $a_j, b_j>0$, and I need to determine whether $A$ is positive definite, meaning that the corresponding quadratic form is bounded below: $$ Q_A(\beta_1, \beta_2\ldots \beta_n\ldots)\stackrel{\mathrm{def}}{=}\sum_{j=1}^\infty a_j \beta_j^2 + 2b_j\beta_{j}\beta_{j+1} \ge c\sum_{j=1}^\infty \beta_j^2.$$ Here $c>0$. (If $c=0$, we say that $A$ is positive semidefinite).

Question Are there infinite-dimensional versions of the familiar criterions of linear algebra, such as the Sylvester's criterion or the diagonal dominance sufficient condition?

Any result or reference is gladly welcome.

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Giuseppe Negro
Giuseppe Negro
  • 717
  • 4
  • 22

I am interested in the following problem: I have an infinite symmetric tridiagonal matrix $$ A_{\infty}= \begin{bmatrix} a_1 & b_1 & & & \\ b_1 & a_2 & b_2 & & \\ & b_2& a_3 & b_3 & \\ & & \ddots & \ddots & \ddots & \\ \end{bmatrix} $$ where $a_j, b_j>0$, and I need to determine whether $A_\infty$ is positive definite, meaning that there exists a constant $c_\infty>0$ such that $$ \sum_{j=1}^\infty a_j \beta_j^2 + \frac{b_j}{2}\beta_{j}\beta_{j+1} \ge c_{\infty}\sum_{j=1}^\infty \beta_j^2$$$$ \sum_{j=1}^\infty a_j \beta_j^2 + 2b_j\beta_{j}\beta_{j+1} \ge c_{\infty}\sum_{j=1}^\infty \beta_j^2$$

Question Are there infinite-dimensional versions of the familiar criterions of linear algebra, such as the Sylvester's criterion or the diagonal dominance sufficient condition?

Any result or reference is gladly welcome.

I am interested in the following problem: I have an infinite symmetric tridiagonal matrix $$ A_{\infty}= \begin{bmatrix} a_1 & b_1 & & & \\ b_1 & a_2 & b_2 & & \\ & b_2& a_3 & b_3 & \\ & & \ddots & \ddots & \ddots & \\ \end{bmatrix} $$ where $a_j, b_j>0$, and I need to determine whether $A_\infty$ is positive definite, meaning that there exists a constant $c_\infty>0$ such that $$ \sum_{j=1}^\infty a_j \beta_j^2 + \frac{b_j}{2}\beta_{j}\beta_{j+1} \ge c_{\infty}\sum_{j=1}^\infty \beta_j^2$$

Question Are there infinite-dimensional versions of the familiar criterions of linear algebra, such as the Sylvester's criterion or the diagonal dominance sufficient condition?

Any result or reference is gladly welcome.

I am interested in the following problem: I have an infinite symmetric tridiagonal matrix $$ A_{\infty}= \begin{bmatrix} a_1 & b_1 & & & \\ b_1 & a_2 & b_2 & & \\ & b_2& a_3 & b_3 & \\ & & \ddots & \ddots & \ddots & \\ \end{bmatrix} $$ where $a_j, b_j>0$, and I need to determine whether $A_\infty$ is positive definite, meaning that there exists a constant $c_\infty>0$ such that $$ \sum_{j=1}^\infty a_j \beta_j^2 + 2b_j\beta_{j}\beta_{j+1} \ge c_{\infty}\sum_{j=1}^\infty \beta_j^2$$

Question Are there infinite-dimensional versions of the familiar criterions of linear algebra, such as the Sylvester's criterion or the diagonal dominance sufficient condition?

Any result or reference is gladly welcome.

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Giuseppe Negro
Giuseppe Negro
  • 717
  • 4
  • 22

Positive definiteness of infinite tridiagonal matrices

I am interested in the following problem: I have an infinite symmetric tridiagonal matrix $$ A_{\infty}= \begin{bmatrix} a_1 & b_1 & & & \\ b_1 & a_2 & b_2 & & \\ & b_2& a_3 & b_3 & \\ & & \ddots & \ddots & \ddots & \\ \end{bmatrix} $$ where $a_j, b_j>0$, and I need to determine whether $A_\infty$ is positive definite, meaning that there exists a constant $c_\infty>0$ such that $$ \sum_{j=1}^\infty a_j \beta_j^2 + \frac{b_j}{2}\beta_{j}\beta_{j+1} \ge c_{\infty}\sum_{j=1}^\infty \beta_j^2$$

Question Are there infinite-dimensional versions of the familiar criterions of linear algebra, such as the Sylvester's criterion or the diagonal dominance sufficient condition?

Any result or reference is gladly welcome.