# Fortunately there are some simple principles from

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Unformatted text preview: 2 2 √ √ − 1 x = 4 0 et + 2 √1 et + 2 2 √2 e9t . 2 1 0 0 Exercise: 2.3.1.a. Find the eigenvalues of the symmetric matrix 5400 4 5 0 0 A= 0 0 4 2 . 0021 b. Find an orthogonal matrix B such that B −1 AB is diagonal. c. Find an orthonormal basis for R 4 consisting of eigenvectors of A. d. Find the general solution to the matrix diﬀerential equation dx = Ax. dt e. Find the solution to the initial value problem 1 3 dx = Ax, x(0) = . 0 dt 2 47 2.3.2.a. Find the eigenvalues of the symmetric matrix −3 2 0 A = 2 −4 2 . 0 2 −5 (Hint: To ﬁnd roots of the cubic, try λ = 1 and λ = −1.) b. Find an orthogonal matrix B such that B −1 AB is diagonal. c. Find an orthonormal basis for R 3 consisting of eigenvectors of A. d. Find the general solution to the matrix diﬀerential equation dx = Ax. dt e. Find the solution to the initial value problem 1 dx = Ax, x(0) = 2 . dt 0 2.3.3. (For students with access to Mathematica) a. Find the eigenvalues of the matrix 211 A = 1 3 2 124 by running the Mathematica program a = {{2,...
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