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Oskar Perron's theorem states that among the latent roots and vectors of a real positive symmetric matrix A there will be a real positive value, to wit the maximum root, which has a positive latent vector (i.e., all components of which are positive, x > 0) and which is not surpassed by any other latent root of the matrix. In a size-shape-time study of a fossil invertebrate species there is a tensorial element involved which operates at different rates in different directions and at typical locations in a tissue. Whenever it is not composed of a vector of ones, the first latent vector includes an expression of shape-variation. Unequal loadings represent different rates of extension in relation to general size. Hence, for a vector with unequal positively signed components, the greater the increase in size, the more will the proportions between the elements diverge. Data on fossil foraminifers and ostracods are used to exemplify the significance of Perron's theorem.

The modified application of Perron's theorem to evolutionary and palaeoecological studies of invertebrates in palaeobiology