# Eigenvalues and Eigenvectors: Properties" In matlib: Matrix Functions for Teaching and Learning Linear Algebra and Multivariate Statistics

(vectors <- ev$vectors)  The eigenvalues are always returned in decreasing order, and each column of vectors corresponds to the elements in values. ## Properties of eigenvalues and eigenvectors The following steps illustrate the main properties of eigenvalues and eigenvectors. We use the notation$A = V' \Lambda V$to express the decomposition of the matrix$A$, where$V$is the matrix of eigenvectors and$\Lambda = diag(\lambda_1, \lambda_2, \dots, \lambda_p)$is the diagonal matrix composed of the ordered eivenvalues,$\lambda_1 \ge \lambda_2 \ge \dots \lambda_p$. 1. Orthogonality: Eigenvectors are always orthogonal,$V' V = I$. zapsmall() is handy for cleaning up tiny values. crossprod(vectors) zapsmall(crossprod(vectors))  1. trace(A) = sum of eigenvalues,$\sum \lambda_i$. library(matlib) # use the matlib package tr(A) sum(values)  1. sum of squares of A = sum of squares of eigenvalues,$\sum \lambda_i^2$. sum(A^2) sum(values^2)  1. determinant = product of eigenvalues,$det(A) = \prod \lambda_i$. This means that the determinant will be zero if any$\lambda_i = 0$. det(A) prod(values)  1. rank = number of non-zero eigenvalues R(A) sum(values != 0)  1. eigenvalues of$A^{-1}$= 1/eigenvalues of A. The eigenvectors are the same, except for order, because eigenvalues are returned in decreasing order. AI <- solve(A) AI eigen(AI)$values
eigen(AI)$vectors  1. There are similar relations for other powers of a matrix: values(mpower(A,p)) = values(A)^p, where mpower(A,2) = A %*% A, etc. eigen(A %*% A) eigen(A %*% A %*% A)$values
eigen(mpower(A, 4))\$values


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matlib documentation built on Oct. 30, 2020, 1:07 a.m.