\documentclass[12pt,twoside]{report}
\usepackage{amsthm,amssymb,amsmath}
\begin{document}
\begin{equation}
\begin{aligned}
&\mathbf{w}(0)=[1,\ 0, \ \ldots, \ 0] \\
&\mathbf{P}(0)=\delta^{-1} \mathbf{I}_{M\times M} \\
& \mathbf{z}(n)=\mathbf{u}(n) \mathbf{u}^{H}(n) \left|\mathbf{u}^{H}(n) \mathbf{w}(n-1)\right|^{p-2} \\
&\mathbf{k}(n)=\frac{\mathbf{P}(n-1) \mathbf{z}(n)}{\lambda+ \mathbf{z}^{H}(n) \mathbf{P}(n-1) \mathbf{z}(n)} \\
&\xi(n)=1- \mathbf{w}^{H}(n-1) \mathbf{z}(n) \\
&\mathbf{w}(n)=\mathbf{w}(n-1)+ \mathbf{k}(n) \xi^{*}(n) \\
&\mathbf{P}(n)=\lambda^{-1} \left(\mathbf{P}(n-1)- \mathbf{k}(n) \mathbf{z}^{H}(n) \mathbf{P}(n-1)\right)
\end{aligned}
\end{equation}
\end{document}