Find the eigenvector and eigenvalues of a 3x3 matrix A using the 3x3 identity matrix.
Eigenvectors and Eigenvalues can be defined as while multiplying a square 3x3 matrix by a 3x1 (column) vector. The result is a 3x1 (column) vector. There are some instances in mathematics and physics where we are interested in which vectors are left "essentially unchanged" by the operation of the matrix and associated constant is called the eigenvalues of the vector v
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.