eigen values

eigen values - Eigenvalues and Eigenvectors, More Direction...

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1 Eigenvalues and Eigenvectors, More Direction Fields and Systems of ODEs First let us speak a bit about eigenvalues. Defn. An eigenvalue λ of an nxn matrix A means a scalar (perhaps a complex number) such that A v = λ v has a solution v which is not the 0 vector. We call such a v an eigenvector of A corresponding to the eigenvalue λ . Note that A v = λ v if and only if 0 = A v - λ v = (A- λ I) v , where I is the nxn identity matrix. Moreover, (A- λ I) v =0 has a non-0 solution v if and only if det(A- λ I)=0. This gives: Theorem. The scalar λ is an eigenvalue of the nxn matrix A if and only if det(A- λ I)=0. Eigenvalue Example. Find the eigenvalues of the matrix By the preceding theorem, we need to solve det(A- λ I)=0. That is 21 . 12 A ⎛⎞ = ⎜⎟ ⎝⎠ 1 0 det 0 1 2 1 det 0. λ ⎩⎭ −− ==
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2 (2+ λ ) 2 -1=0 4+4 λ + λ 2 -1=0 3+4 λ + λ 2 =0 (3+ λ )(1+ λ )=0 λ =-1,-3. These are the eigenvalues of A. You can use Gaussian elimination to find the corresponding eigenvectors. (A+3I) v = 0 is solved as follows: 21 3 0 1 1 3. 12 0 3 1 1 AI ⎛⎞ += + = ⎜⎟ ⎝⎠ To find v , do the Gauss thing: 11 0 1 1 0 . 0 0 0 0 Here we replaced row 2 by row 2 – row 1. Now a solution v of (A+3I) v =0 means 1 2 0. v v such that v v v =+ = So v 1 =-v 2 . Take v 2 =-1 (the free unknown coefficient) and then v 1 =1. An eigenvector of A for the eigenvalue -3 is Any scalar multiple of v is also an eigenvector. 1 . 1 v =
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3 Similarly we find a solution w of (A+I) w =0.
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eigen values - Eigenvalues and Eigenvectors, More Direction...

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