is.ergodic: 2. TEST ERGODICITY OF A MATRIX
In akeyel/spatialdemography: A Spatially Explicit Metacommunity Model

Description Usage Arguments Author(s)

This function will test the ergodicity of a matrix based upon whether its dominant left eigenvector is positive or not. A reducible-ergodic matrix has a positive dominant left eigenvector, where a reducible-nonergodic matrix will contain zeroes in its dominant left eigenvector. An irreducible matrix is always ergodic. NOTE: this function can prove problematic for imprimitive, irreducible matrices or reducible matrices with imprimitive, irreducible blocks on the diagonal. The 'dominant' left eigenvector chosen by R is that with the largest absolute value (including both real and imaginary components), so that sometimes it chooses a dominant with imaginary components. In addition, where a reducible matrix in block-permuted form contains one or more irreducible, imprimitive blocks on the diagonal, values of zero in the dominant left eigenvector may actually be calculated by R as small numbers, approximate to zero. Hence, for such matrices, R may return a spurious result of the matrix being ergodic when in fact it is nonergodic. For this reason, the 'dominant' left eigenvector is also returned by the function so that it may be checked by eye: if it contains non-zero imaginary components, or if it contains small numbers that may approximate to zero then further diagnosis may be required. The 'blockmatrix' and 'is.primitive' functions can help to diagnose whether diagonal blocks are imprimitive or not.