How large can the off-diagonal be?

MediumLinear Algebra~4m

The symmetric matrix $\begin{pmatrix} 5 & b \\ b & 2 \end{pmatrix}$ is positive definite exactly when $|b| < b^*$ for some threshold $b^*$ . Find $b^*$ . (to 4 decimals)

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Sylvester: a symmetric matrix is PD iff all leading principal minors are positive.
The binding one is $\det = 10 - b^2 > 0$ , so $|b| < \sqrt{10}$ .

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3.1623 (± 0.001)

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How large can the off-diagonal be?

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