Description Usage Arguments Details Value Author(s) References Examples

Finds a dominant eigenvalue, *λ_1*, and its corresponding
eigenvector, *v_1*, of a square matrix by applying Hotelling's (1933) Power Method with scaling.

1 | ```
powerMethod(A, v = NULL, eps = 1e-06, maxiter = 100, plot = FALSE)
``` |

`A` |
a square numeric matrix |

`v` |
optional starting vector; if not supplied, it uses a unit vector of length equal to the number of rows / columns of |

`eps` |
convergence threshold for terminating iterations |

`maxiter` |
maximum number of iterations |

`plot` |
logical; if |

The method is based upon the fact that repeated multiplication of a matrix *A* by a trial
vector *v_1^{(k)}* converges to the value of the eigenvector,

*v_1^{(k+1)} = A v_1^{(k)} / \vert\vert A v_1^{(k)} \vert\vert *

The corresponding eigenvalue is then found as

*λ_1 = \frac{v_1^T A v_1}{v_1^T v_1}*

In pre-computer days, this method could be extended to find subsequent eigenvalue - eigenvector
pairs by "deflation", i.e., by applying the method again to the new matrix.
*A - λ_1 v_1 v_1^{T} *.

This method is still used in some computer-intensive applications with huge matrices where only the dominant eigenvector is required, e.g., the Google Page Rank algorithm.

a list containing the eigenvector (`vector`

), eigenvalue (`value`

), iterations (`iter`

),
and iteration history (`vector_iterations`

)

Gaston Sanchez (from matrixkit)

Hotelling, H. (1933). Analysis of a complex of statistical variables into principal components. *Journal of Educational Psychology*, 24, 417-441, and 498-520.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 | ```
A <- cbind(c(7, 3), c(3, 6))
powerMethod(A)
eigen(A)$values[1] # check
eigen(A)$vectors[,1]
# demonstrate how the power method converges to a solution
powerMethod(A, v = c(-.5, 1), plot = TRUE)
B <- cbind(c(1, 2, 0), c(2, 1, 3), c(0, 3, 1))
(rv <- powerMethod(B))
# deflate to find 2nd latent vector
l <- rv$value
v <- c(rv$vector)
B1 <- B - l * outer(v, v)
powerMethod(B1)
eigen(B)$vectors # check
# a positive, semi-definite matrix, with eigenvalues 12, 6, 0
C <- matrix(c(7, 4, 1, 4, 4, 4, 1, 4, 7), 3, 3)
eigen(C)$vectors
powerMethod(C)
``` |

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