# Iterative Matrix Techniques Based on Averages
## 2 Averaged Power Method for Finding Eigenvalues and Eigenvectors of a Matrix  
Given a matrix \(A\in\mathcal{M}(m\times m)\), where \(\mathcal{M}(m\times m)\) denotes the set of real or complex matrices of size \(m\times m\), the power method is a well-known procedure for finding the dominant eigenvalue of matrix \(A\). Let \(\sigma(A)\) represent the spectrum of \(A\), that is to say, the set of eigenvalues of \(A\). Definition 1: A matrix \(A\) has a dominant eigenvalue \(\lambda_{1}\in\sigma(A)\) if \(|\lambda_{1}|>|\lambda_{j}|\) for any \(\lambda_{j}\in\sigma(A)\), \(j\
eq 1\). If \(v\in\mathbb{R}^{m}\) (or \(v\in\mathbb{C}^{m}\)) is non-null and \(Av=\lambda_{1}v\) then \(v\) is a dominant eigenvector of \(A\). It is well known that if the matrix \(A\) is diagonalizable and it has a dominant eigenvalue, then the sequence defined by \(X_{0}\
eq\bar{0}\) and the recurrence \(X_{k+1}=AX_{k}\) for \(k\geq 0\) converges with high probability to a dominant eigenvector (i.e., if the component of \(X_{0}\) with respect to the dominant eigenvector is non-null). The dominant eigenvalue can be approximated by means of the so-called Rayleigh quotient:  
\[\lambda_{1}\simeq\frac{<AX_{k},X_{k}>}{<X_{k},X_{k}>},\]  
when \(k\) is sufficiently large. The formula comes, obviously, from the definition of an eigenvector associated with the eigenvalue \(\lambda_{1}\). We shall see, with an example, that if the matrix has no dominant eigenvalue then the power method may not converge. Example 1: The matrix \(A=diag(1,-1)\) has the eigenvalue \(\lambda_{1}=1\) with eigenvectors \((x,0)\) and \(\lambda_{2}=-1\) with eigenvectors \((0,y)\). The power method does not work in general since  
\[A^{k}(x,y)=(x,-y),\]  
for \(k\) being odd and  
\[A^{k}(x,y)=(x,y)\]  
for \(k\) being even. The sequence \((A^{k}X_{0})\) does not converge except for the starting point \(X_{0}=(x,0)\)The present section is devoted to proposing an algorithm to find eigenvalues and eigenvectors of matrices that do not have a dominant eigenvalue. In reference [22], an algorithm for the approximation of a fixed point of \(T:C\to C\), where \(C\), is a nonempty, closed and convex subset of a proposed normed space \(E\). This is given by the following recurrence:  
\[g_{n} =(1-\alpha_{n})f_{n}+\alpha_{n}Tf_{n},\] \[h_{n} =(1-\beta_{n})f_{n}+\beta_{n}g_{n},\] \[f_{n+1} =(1-\gamma_{n})h_{n}+\gamma_{n}Th_{n},\]  
for \(\alpha_{n},\beta_{n},\gamma_{n}\in[0,1]\), \(n\geq 0\) and \(f_{0}\in C\). This iterative scheme was called N-algorithm. The procedure generalizes the Krasnoselskii-Mann method, defined in references [6; 7], obtained in the case where \(\alpha_{n}=0,\beta_{n}=1\). If, additionally, \(\gamma_{n}=1\) one has the usual Picard-Banach iteration. **Averaged power method**  
If the matrix \(A\in\mathcal{M}(m\times m)\) has no dominant eigenvalue, that is to say, \(|\lambda_{j}|=k\
eq 0\) for any \(\lambda_{j}\in\sigma(A)\), the power method may not converge. The algorithm proposed here is the use of the N-iteration of the matrix \(A\) instead of computing the powers of \(A\). The sequence of vectors obtained by the power or the averaged power method may produce a divergent sequence (even though the iterates may be good approximations of an eigenvector). To avoid this fact, a scaling of the output vector is sometimes performed at every step of the algorithm. Thus, the procedure is described by the following recurrence:  
\[V_{n}=(1-\alpha_{n})X_{n}+\alpha_{n}AX_{n},\] \[W_{n}=(1-\beta_{n})X_{n}+\beta_{n}V_{n},\] \[X_{n+1}=(1-\gamma_{n})W_{n}+\gamma_{n}AW_{n},\] \[X_{n+1}:=X_{n+1}/||X_{n+1}||,\] \[A^{k}(x,y)=(x,-y),\]0  
for \(\alpha_{n},\beta_{n},\gamma_{n}\in[0,1]\), such that \(0<\inf\alpha_{n}\leq\sup\alpha_{n}<1\), \(0<\inf\gamma_{n}\leq\sup\gamma_{n}<1\), and \(X_{0}\
eq 0\), \(X_{0}\in\mathbb{R}^{m}\). The notation \(||\cdot||\) represents any norm in \(\mathbb{R}^{m}\) and \(<\cdot,\cdot>\) is the inner product in the \(m\)-dimensional Euclidean space.