pf_eigenvalue function

Perron-Frobenius Eigenvalue

The Perron-Frobenius eigenvalue of a positive matrix. methods

pf_eigenvalue(X)

Arguments

• X: An Expression or positive square matrix.

Returns

An Expression representing the largest eigenvalue of the input.

Details

For an elementwise positive matrix $X$, this atom represents its spectral radius, i.e., the magnitude of its largest eigenvalue. Because $X$ is positive, the spectral radius equals its largest eigenvalue, which is guaranteed to be positive.

This atom is log-log convex.

Examples

n <- 3
X <- Variable(n, n, pos=TRUE)
objective_fn <- pf_eigenvalue(X)
constraints <- list( X[1,1]== 1.0,
                     X[1,3] == 1.9,
                     X[2,2] == .8,
                     X[3,1] == 3.2,
                     X[3,2] == 5.9,
                     X[1, 2] * X[2, 1] * X[2,3] * X[3,3] == 1)
problem <- Problem(Minimize(objective_fn), constraints)
result <- solve(problem, gp=TRUE)
result$value
result$getValue(X)