Eigencomponents of symmetric matrices

If A is a symmetric matrix, then the matrix V is orthogonal. This can be shown by considering two eigencomponent pairs i and j 

If A is a symmetric matrix, it can be shown that its eigenvalues are real. Product-matrices such as ATA or AAT are special cases of symmetric matrices and will be shown in the next section to have non-negative eigenvalues. 

The elements of A1/2 may be complex numbers. A1/2 is the product of a simple rotation and a scaling, carried out in that order. 

Multiplication of vector x by matrix A produces the vector y and therefore 

Using the results from the previous exercise, the first transformation UTx is a rotation that can be written as 

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