Quiz VII1. The matrixA=⎛⎝201010102⎞⎠is symmetric. Find a basis forR3consisting ofeigenvectors forA.Solution:SinceAis symmetric,there will beanorthonormalbasis consisting ofeigenvectors ofA, but you do not need to use that to solve the problem. First ±ndthe eigenvaluesdet⎛⎝2−λ01−λ0102−λ⎞⎠=(2−λ)2(1−λ)−(1−λ)=(1−λ)(3−4λ+λ2).So the eigenvalues are 3 and 1. Solving for the eigenspaces,±v1=⎛⎝101⎞⎠is aneigenvector for the eigenvalue 3. For the eigenvalue 1,A−I=⎛⎝101000⎞⎠,and the reduced row echelon form of that is just
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This note was uploaded on 04/12/2011 for the course MATH 33a taught by Professor Lee during the Fall '08 term at UCLA.