Calculating Eigencomponents

This is non-trivial. We can see this by noting that the property MCj = A y-6, -can be rewritten as 

The matrix [M — Ajl] must be of reduced rank to give a zero result when it multiplies a non-zero vector, and it therefore has a zero determinant. 

Thus Xj is a root of the equation, polynomial in A, det(M — XI) = 0 and computing the eigenvalues is equivalent to finding the roots of that polynomial, which is called the characteristic polynomial. 

Galois proved that this is non-trivial. If the size of M is greater than 4x4, then there is no algebraic closed form, and if the size of M is greater than 2x2 there is no practical closed form. 

Any algorithm for computing eigenvalues must therefore be iterative. The iteration might be hidden inside a polynomial solver or explicit (as in the QR algorithm) but it will always be there, unless some other information about the matrix can be exploited. 

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Why do all the arithmetic of calculating eigencomponents, when there is software available to do it for us ... 

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