{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Eigenvector_Eigenvalue Method

Eigenvector_Eigenvalue Method - The Eigenvector/Eigenvalue...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
The Eigenvector/Eigenvalue Method We first recall the following from linear algebra: Definition Let A be an n × n matrix. A vector v such that A v = λ v for some scalar λ is called an eigenvector (EV) of A with eigenvalue (ev) λ . Before we discuss how these objects can be applied to solving the homogeneous equation ˙ y = A y , we review the process for finding the eigenvectors and eigenvalues of a given matrix A . Note that Av = λ v m ( A - λI ) v = 0 det( A - λI ) = 0 where I is the n × n identity matrix (the matrix such that every entry on the main diagonal is 1 and every other entry is zero). Since det( A - λI ) is a polynomial in the variable λ , we need only find the zeros of this polynomial to find all the eigenvalues of A . To find a corresponding eigenvector, we must solve the system ( A - λI ) v = 0 for the components of v . I Example Find the eigenvalues and corresponding eigenvectors of the matrix A = 4 2 - 1 1 . Solution The eigenvalues are the roots of the polynomial det( A - λI ) . We compute A - λI = 4 2 - 1 1 - λ 1 0 0 1 = 4 - λ 2 - 1 1 - λ Then we compute det( A - λI ) = (4 - λ )(1 - λ ) - ( - 1)(2) = λ 2 - 5 λ + 6 Setting this to zero we have 0 = λ 2 - 5 λ + 6 = ( λ - 2)( λ - 3) so the eigenvalues are λ 1 = 2 and λ 2 = 3 .
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
To find the eigenvectors v = v 1 v 2 corresponding to λ 1 = 2 , we need to solve ( A - λ 1 I ) v = 0 that is, 4 2 - 1 1 - 2 1 0 0 1 v 1 v 2 = 0 0 . 2 2 - 1 - 1 v 1 v 2 = 0 0 2 v 1 + 2 v 2 = 0 - v 1 - v 2 = 0 Since the second equation is just a scalar multiple of the first, this system has infinitely many solutions of the form v 1 = - v 2 . Then the eigenvectors corresponding to λ 1 = 2 are all of the form v = c - c = c 1 - 1 where c is arbitrary.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}