Diagonalization Calculator

P converts coordinates from the standard basis to the eigenvector basis. D applies simple scaling along each eigenvector direction. P inverse converts back. Together they reproduce the same transformation as A but through a much simpler path.

Because diagonal matrices are trivial to compute with. Need A to the 100th power? Instead of 100 matrix multiplications, compute P diag(lambda1^100, lambda2^100) P inverse, which is just one multiplication. This extends to matrix exponentials, differential equations, and Markov chains.

No. A 2x2 matrix cannot be diagonalized if it has a repeated eigenvalue with only a 1-dimensional eigenspace (like [[2, 1], [0, 2]]), or if it has complex eigenvalues and you need real diagonalization.

D is a diagonal matrix with the eigenvalues on the diagonal. The order of eigenvalues in D must match the order of the corresponding eigenvectors as columns of P.

Compute the product PDP inverse and check that it equals the original matrix A. You can also verify by checking that each column of P is an eigenvector: A times column i of P should equal lambda i times column i of P.

A system dx/dt = Ax can be decoupled by diagonalization. In the eigenvector basis, the system becomes dy/dt = Dy, which is just independent exponential equations. The solution is y(t) = e^(Dt) y(0), then convert back with x = Py.

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