ABSTRACT
Among the easiest matrices to deal with are the diagonal matrices, those in which all terms not on the main diagonal are 0. It is reasonable to ask which linear transformations T from a vector space V to itself have diagonal matrices with respect to some basis. If T has such a basis and v → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315138695/b832c43e-68c9-4052-aad2-0d41b672a31d/content/eq5876.tif"/> is a basis vector, then T v → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315138695/b832c43e-68c9-4052-aad2-0d41b672a31d/content/eq5877.tif"/> is a multiple of v → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315138695/b832c43e-68c9-4052-aad2-0d41b672a31d/content/eq5878.tif"/> . Thus we are led to study vectors v → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315138695/b832c43e-68c9-4052-aad2-0d41b672a31d/content/eq5879.tif"/> which are multiplied by some constant λ under the action of T. These vectors, called eigenvectors, and the constants that arise, called eigenvalues, are of considerable importance in many situations.
