Question
Find two positive definite matrices $K$ and $L$ whose product $K L$ is not positive definite.
Step 1
A matrix $A$ is positive definite if it is symmetric ($A = A^T$) and all its eigenvalues are positive. This implies that for any non-zero vector $x$, $x^T A x > 0$. Show more…
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Find the values of $k$ that make each of the following matrices positive definite: (a) $A=\left[\begin{array}{rr}2 & -4 \\ -4 & k\end{array}\right]$ (b) $\quad B=\left[\begin{array}{ll}4 & k \\ k & 9\end{array}\right]$ (c) $C=\left[\begin{array}{rr}k & 5 \\ 5 & -2\end{array}\right]$ (a) First, $k$ must be positive. Also, $|A|=2 k-16$ must be positive; that is, $2 k-16>0 .$ Hence, $k>8$ (b) We need $|B|=36-k^{2}$ positive; that is, $36-k^{2}> 0 .$ Hence, $k^{2}<36$ or $-6< k< 6$ (c) $C$ can never be positive definite, because $C$ has a negative diagonal entry -2.
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