# Decompose symmetric matrix to scaling factors

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 $Λ$)?

If you know $Z$ and $P$, then by comparing $z_{ii}$ and $p_{ii}$ (unless they are $0$), you can find $\lambda_i^2$...
• ugh, of course. i was simply confused by the redundancy. $λ_i = \sqrt{p_{ii} ÷ z_{ii}}$ – flying sheep Dec 13 '13 at 14:08