Eigen vector, Eigen value 3x3 Matrix Calculator

A vector v for which this equation hold is called an eigenvector of the matrix A and the associated constant is called the eigenvalues of the vector v.

Eigenvectors-Eigenvalues can be defined as while multiplying a square 3x3 matrix by a 3x1 (column) vector. The result is a 3x1 (column) vector. The 3x3 matrix can be thought of as an operator - it takes a vector, operates on it, and returns a new vector. There are some instances in mathematics and physics in which we are interested in which vectors are left "essentially unchanged" by the operation of the matrix. Specifically, we are interested in those vectors v for which Av=kv where A is a square matrix and k is a real number. A vector v for which this equation hold is called an eigenvector of the matrix A and the associated constant k is called the eigenvalues of the vector v. If a matrix has more than one eigenvector the associated eigenvalues can be different for the different eigenvectors.In geometry, the action of a matrix on one of its eigenvectors causes the vector to shrink/stretch and/or reverse direction.

In order to find the eigenvalues of a 3x3 matrix A, we solve Av=kv for scalar(s) k. Rearranging, we have Av-kv=0. But kv=kIv where I is the 3x3 identity matrix

Applications of eigenvalues and eigenvectors

Eigenvectors make understanding linear transformations. Eigenvalues and eigenvectors allow us to "reduce" a linear operation to separate, simpler, problems. For example if a stress is applied to a "plastic" solid, the deformation can be dissected into "principle directions"- those directions in which the deformation is greatest.