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matrixeigenexample

# matrixeigenexample - Example of a Symmetric Matrix with...

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Page 1 of 3 1/28/2008 matrixeigenexample.doc >> A=[ 5 4 1 1; 4 5 1 1;1 1 4 2;1 1 2 4] A = 5 4 1 1 4 5 1 1 1 1 4 2 1 1 2 4 >> [X,LAM]=eig(A) X = 0.7071 -0.0000 0.3162 0.6325 -0.7071 -0.0000 0.3162 0.6325 0.0000 0.7071 -0.6325 0.3162 -0.0000 -0.7071 -0.6325 0.3162 LAM = 1.0000 0 0 0 0 2.0000 0 0 0 0 5.0000 0 0 0 0 10.0000 >> X'*X ans = 1.0000 0.0000 -0.0000 -0.0000 0.0000 1.0000 -0.0000 0 -0.0000 -0.0000 1.0000 0.0000 -0.0000 0 0.0000 1.0000 >> X*LAM*X' ans = 5.0000 4.0000 1.0000 1.0000 4.0000 5.0000 1.0000 1.0000 1.0000 1.0000 4.0000 2.0000 1.0000 1.0000 2.0000 4.0000 >> A*X-X*LAM ans = 1.0e-014 * 0 -0.0236 -0.1110 0.0888 -0.0555 -0.0340 -0.1776 0.0888 0.0382 0.0888 -0.0888 0.2220 0.0075 0.1110 0.0444 0.1776 Set up this 4 by 4 symmetric matrix Use the Matlab eig function to compute the X matrix and the LAM matrix. The columns of X are the eigenvectors. The diagonals of LAM are the corresponding eigenvalues. The X matrix is orthogonal: i.e. its transpose is its inverse. The matrix can be represented as A = X*LAM*X’ Note that A*X = X*LAM since A*X- X*LAM is zero with the numerical tolerance of Matlab. It is easy to show that the eigenvectors can be scaled so that they have integer values this is unusual to have all integer values, but it makes this special A matrix useful in learning the basis of the symmetric eigenvalue problem.

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