Shear mapping in the context of Eigenvalues

In mathematics, a square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order ${\displaystyle n}$. Any two square matrices of the same order can be added and multiplied.

Square matrices are often used to represent simple linear transformations, such as shearing or rotation. For example, if ${\displaystyle R}$ is a square matrix representing a rotation (rotation matrix) and ${\displaystyle \mathbf {v} }$ is a column vector describing the position of a point in space, the product ${\displaystyle R\mathbf {v} }$ yields another column vector describing the position of that point after that rotation. If ${\displaystyle \mathbf {v} }$ is a row vector, the same transformation can be obtained using ${\displaystyle \mathbf {v} R^{\mathsf {T}}}$, where ${\displaystyle R^{\mathsf {T}}}$ is the transpose of ${\displaystyle R}$.

In linear algebra, an eigenvector (/ˈaɪɡən-/ EYE-gən-) or characteristic vector is a (nonzero) vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector ${\displaystyle \mathbf {v} }$ of a linear transformation ${\displaystyle T}$ is scaled by a constant factor ${\displaystyle \lambda }$ when the linear transformation is applied to it: ${\displaystyle T\mathbf {v} =\lambda \mathbf {v} }$. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor ${\displaystyle \lambda }$ (possibly a negative or complex number).

Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. A linear transformation's eigenvectors are those vectors that are only stretched or shrunk, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or shrunk. If the eigenvalue is negative, the eigenvector's direction is reversed.

Biological or process structuralism is a school of biological thought that objects to an exclusively Darwinian or adaptationist explanation of natural selection such as is described in the 20th century's modern synthesis. It proposes instead that evolution is guided differently, by physical forces which shape the development of an animal's body, and sometimes implies that these forces supersede selection altogether.

Structuralists have proposed different mechanisms that might have guided the formation of body plans. Before Darwin, Étienne Geoffroy Saint-Hilaire argued that animals shared homologous parts, and that if one was enlarged, the others would be reduced in compensation. After Darwin, D'Arcy Thompson hinted at vitalism and offered geometric explanations in his classic 1917 book On Growth and Form. Adolf Seilacher suggested mechanical inflation for "pneu" structures in Ediacaran biota fossils such as Dickinsonia. Günter P. Wagner argued for developmental bias, structural constraints on embryonic development. Stuart Kauffman favoured self-organisation, the idea that complex structure emerges holistically and spontaneously from the dynamic interaction of all parts of an organism. Michael Denton argued for laws of form by which Platonic universals or "Types" are self-organised. Stephen J. Gould and Richard Lewontin proposed biological "spandrels", features created as a byproduct of the adaptation of nearby structures. Gerd B. Müller and Stuart A. Newman argued that the appearance in the fossil record of most of the phyla in the Cambrian explosion was "pre-Mendelian" evolution caused by physical factors. Brian Goodwin, described by Wagner as part of "a fringe movement in evolutionary biology", denies that biological complexity can be reduced to natural selection, and argues that pattern formation is driven by morphogenetic fields.