MIT18_03S10_c33

MIT18_03S10_c33 - 18.03 Class 33, Apr 28, 2010 Eigenvalues...

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18.03 Class 33 , Apr 28, 2010 Eigenvalues and eigenvectors 1. Linear algebra 2. Ray solutions 3. Eigenvalues 4. Eigenvectors 5. Initial values [1] Prologue on Linear Algebra. Recall [a b ; c d] [x ; y] = x[a ; c] + y[b ; d] : A matrix times a column vector is the linear combination of the columns of the matrix weighted by the entries in the column vector. When is this product zero? One way is for x = 0 = y. If [a ; c] and [b ; d] point in different directions, this is the ONLY way. But if they lie along a single line, we can find x and y so that the sum cancels. Write A = [a b ; c d] and u = [x ; y] , so we have been thinking about A u = 0 as an equation for u . It always has the "trivial" solution u = 0 = [0 ; 0] : 0 is a linear combination of the two columns in a "trivial" way, with 0 coefficients, and we are asking when it is a linear combination of them in a different, "nontrivial" way. We get a nonzero solution [x ; y] exactly when the slopes of the vectors [a ; c] and [b ; d] coincide: c/a = d/b , or ad - bc = 0. This combination of the entries in A is so important that it's called the "determinant" of the matrix: det(A) = ad - bc We have found: Theorem: Au = 0 has a nontrivial solution exactly when det A = 0 . If A is a larger *square* matrix the same theorem still holds, with the appropriate definition of the number det A . [2] Solve u' = Au : for example with A = [-2 1 ; -4 3] . The Mathlet "Linear Phase Portraits: Matrix entry" shows that some
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MIT18_03S10_c33 - 18.03 Class 33, Apr 28, 2010 Eigenvalues...

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