MIT
18 06 -
Lecture 27
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Lecture 27 When matrices are symmetric A is equal to A transpose The eigenvalues are real We know Ax x so we can write it in terms of its complex conjugate Ax x Now we transpose the matrix to get and then we use the assumption that A A Using the original equation and multiplying it by x we get Finally we get that and this means that all of the eigenvalues are real The eigenvectors are orthogonal Usually we write that A S S but now we can write that as A Q Q Where Q has the orthonormal eigenvectors in its columns Every symmetric matrix is a combination of perpendicular projection matrices For symmetric matrices the sign of the pivots is the same as the sign of the eigenvalues
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