Eigenvalues and Eigenvectors in Geology

In a square matrix, eigenvectors are usually represented by the non-zero vectors that when multiplied by the matrix; remain parallel to the initial vector. The corresponding eigenvalue for each eigenvector is the factor with which the eigenvector is scaled whenever multiplied by the matrix. Eigen values/vectors are widely used in various areas including physics, biology, economics, sociology, engineering, geology, and statistics. Engineers mainly use Eigen values/vectors to reduce noise in cars, vibration analysis, structural analysis, stereo systems, and material analysis (Trigg 36).

In geology, particularly when geologists are studying glacial till, Eigen values/vectors assist in summarization of the information of the clasts constituentsâ€™ orientation/dip into a 3-D Space. A geologist first collects the data for various clasts in the selected soil sample and later on the data is compared graphically using a Steronet on a Wulff Net or a Tri-Plot diagram. The output of the orientation tensor is usually in three perpendicular axes of space, which have three eigenvectors ordered v1, v2, and v3. Their corresponding Eigen values include E1, E2, and E3, in which E1>E2>E3 (Cs.mcgill.ca 1).

In terms of strength, v1 is classified as the primary orientation/dip, while v2 as the secondary, and v3 the tertiary orientation/dip. The clasts orientation is got by calculating the direction of the corresponding eigenvector. If all the Eigen values are equal (E1=E2=E3), then it is concluded that the fabric is isotropic. If the first two Eigen values are equal, and greater than the third value (E1=E2>E3), the fabric is planar. In a situation whereby E1>E2>E3, the fabric is considered to be linear (Wikipedia 1).

References

Cs.mcgill.ca,. ‘Eigenvalue, Eigenvector And Eigenspace’. N.p., 2015. Web. 1 July 2015. http://www.cs.mcgill.ca/~rwest/link-suggestion/wp01cd_2008-09_augmented/wp01/e/Eigenvalue%252C_eigenvector_and_eigenspace.htm

Trigg, George L. Mathematical Tools for Physicists.Weinheim: Wiley-VCH, 2005. Internet resource.

Wikipedia,. ‘Eigenvalues and Eigenvectors’. N.p., 2015. Web. 1 July 2015. https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors#Applications