[Math] A question about the positive definite matrices and condition number

matrices

Let $A=LDL^T$ be a symmetric positive definite matrix, where $L$ is a unit lower triangular matrix, and $D=\textrm{diag}(d_{ii}).$
Show that $$\textrm{Cond}_2(A) \geq \frac{\max (d_{ii})}{\min (d_{ii})}.$$

Best Answer

Hint: note the following:

  • $\textrm{Cond}_2(A) = \|A\| \|A^{-1}\|$
  • Because $A$ is positive semidefinite, $\|A\| = \max_{\|x\| = 1} x^TAx$
  • Because $A$ is positive semidefinite, $\|A^{-1}\| = \max_{\|x\| = 1} \frac{1}{x^TAx}$

For an lower bound of each of these maxima, plug in the $j$th standard basis vector for $x$.

Let $L_j$ denote the $j$th row of $L$. We have $$ e_j^T(LDL^T)e_j = d_{jj}\cdot \|L_j\|^2 \geq d_{jj} $$

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