Eigenvector decomposition. Equivalent forms.

The above is equivalent to the more commonly seen:

And another equivalent is:

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

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:

Is simply $Av=\lambda v$, but for all eigenvectors at once. As $\mathrm{\Lambda }$ is a diagonal matrix, we can say:

which is slighly closer to the form $Av=\lambda v$.