Ex 10 - The minimum and maximum eigenvalues of matrix are –2 and 6 respectively. The other two

GATE Eigenvalue Eigenvector Problems Short Cut Method Example 10

Q. The minimum and maximum eigen values of matrix are –2 and 6 respectively. The other two eigen values are (GATE CE 2007)

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.

This part of the lecture covers short cut method to solve eigenvalue and eigenvector problems commonly asked in GATE exam.

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