ED01
Eigendeomposition Theorem Example 1
The eigenvalues and the correspondig eigenvectors of a 2x2 matrix are given by
Theory
Eigenvalue problems often show up in physics and many branches of engineering. Such problems are in the form
A v = lambda * v
Here A is a matrix, v is eigenvector, lambda is eigenvalue. Eigenvector and eigenvalue are often incorrectly written as eigen vector and eigen value.
The matrix A corresponds to some linear transformation on vector v. The vector v satisfying above equation is called eigenvector.
In general when we multiply a matrix A ( n x n ) with a vector v (n x 1) we get a vector Av, which is not parallel to v. However, when v is an eigenvector of A then A v = lambda v is parallel to v. lambda is the factor by which v is scaled, i.e., magnitude of v is multiplied by lambda. lambda is called the eigenvalue. v is eigenvector corresponding to eigenvalue lambda .
In GATE most of eigenvalue / eigenvector problems are solved using properties of eigenvalues and eigenvectors.
Property 1. When a matrix is multiplied by its eigenvector then the eigenvector gets scaled by eigenvalue corresponding to the vector without change in the direction ( rotation ) of the eigenvector.
Property 2. Sum of eigenvalues of a matrix is equal to trace of the matrix, i.e., sum of diagonal elements of the matrix.
Property 3. Product of eigenvalues of a matrix is equal to determinant of the matrix.
Eigendecomposition Theorem
A rectangular matrix can always be written as A = SDS-1 here columns of matrix S are eigenvectors of A and D is a diagonal matrix with corresponding eigenvalues as the diagonal elements.
This part of the lecture covers short cut method to solve eigenvalue and eigenvector problems commonly asked in GATE exam.
This video is uploaded by
Alpha Academy, Udaipur
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Minakshi Porwal (9460189461)
