4
$\begingroup$

I have a symmetric square matrix $P$ composed by left- and right-multiplying another symmetric square matrix $Z$ with a diagonal matrix $Λ$:

$$P = ΛZΛ$$

i.e. ($λ_i$ means $λ_{ii}$):

$$ \begin{bmatrix} p_{11} & p_{12} & \cdots & p_{1n} \\ p_{21} & p_{22} & \cdots & p_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ p_{n1} & p_{n2} & \cdots & p_{nn} \\ \end{bmatrix} = \begin{bmatrix} λ_1λ_1z_{11} & λ_1λ_2z_{12} & \cdots & λ_1λ_nz_{1n} \\ λ_2λ_1z_{21} & λ_2λ_2z_{22} & \cdots & λ_2λ_nz_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ λ_nλ_1z_{n1} & λ_nλ_2z_{n2} & \cdots & λ_nλ_nz_{nn} \\ \end{bmatrix} $$

Is there a decomposition able to do the reverse operation (going from $Z$ and $P$ to $Λ$)?

$\endgroup$
3
$\begingroup$

If you know $Z$ and $P$, then by comparing $z_{ii}$ and $p_{ii}$ (unless they are $0$), you can find $\lambda_i^2$...

$\endgroup$
  • $\begingroup$ ugh, of course. i was simply confused by the redundancy. $λ_i = \sqrt{p_{ii} ÷ z_{ii}}$ $\endgroup$ – flying sheep Dec 13 '13 at 14:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.