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Math and science::Algebra

Eigenvector decomposition. Equivalent forms.

The following is true, for a diagonizable matrix \( A \):

\[ S^{-1} A S = \Lambda \]

The above is equivalent to the more commonly seen:

{{\[ A = S \Lambda S^{-1} \]

And another equivalent is:

\[ AS = S \Lambda \]

Adding brackets to emphasize a multiplication order can help make the expressions more intuitive:

\[ S^{-1} (A (S)) = \Lambda \]

involves taking \( A \) times it's eigenvectors, which will produce scaled versions of these eigenvectors, and then expressing these scaled columns in the basis of the eigenvectors, which of course will just be a single number (an eigenvalue) for each column.

The expression:

\[ A S = S \Lambda \]

Is simply \( A v = \lambda v \), but for all eigenvectors at once. As \( \Lambda \) is a diagonal matrix, we can say:

\[ A S = S \Lambda = \Lambda S \]

which is slighly closer to the form \( A v = \lambda v \).