Computation of Eigenvectors An eigenvector of a linear transformation T is a nonzero vector v such that T(v) = λv for some scalar value λ . In...
An eigenvector of a linear transformation T is a nonzero vector v such that T(v) = λv for some scalar value λ. In other words, v is an eigenvector if it is stretched or shrunk by T in a way that is proportional to the eigenvalue.
Eigenvalues are the non-zero scalar values associated with eigenvectors. They tell us how the eigenvectors transform under T, and they are used to solve problems involving linear transformations.
To compute the eigenvectors of a linear transformation, we need to find the eigenvalues and corresponding eigenvectors of the corresponding linear operator.
Finding Eigenvalues:
We consider the characteristic polynomial of T, which is a polynomial p(x) whose roots are the eigenvalues.
The eigenvalues are the roots of p(x).
Finding Eigenvectors:
For each eigenvalue λ, we solve the equation (T - λI)v = 0, where I is the identity matrix.
The solutions to this equation form a set of eigenvectors corresponding to the eigenvalue.
Each eigenvector is a nonzero vector that is transformed into a multiple of itself under T.
The eigenvectors are found by solving a system of linear equations involving T and the eigenvector.
Examples:
Let T be the transformation that rotates the plane counterclockwise by 45 degrees.
The characteristic polynomial of T is (x^2 - 1).
The eigenvalues of T are 1 and -1.
The eigenvectors of T corresponding to the eigenvalue 1 are v1 = (1, 0) and v2 = (0, 1).
The eigenvectors corresponding to the eigenvalue -1 are v3 = (-1, 0) and v4 = (0, 1).
In conclusion, understanding eigenvectors and eigenvalues is crucial for comprehending the behavior of linear transformations and solving problems involving them