Find all invariant subspace of $A$ Consider a matrix mapping $A: V \to V$ for a vertor space V. Matrix $A$ has 3 eigenvalues are distinct : $\lambda_1,\lambda_2,\lambda_3$ and $v_1,v_2,v_3$ are vectors-corresponding .Find all invariant subspace of  $A$

The question of finding all $A$-invariant subspaces is rather difficult in general. However it has a fairly easy answer in the case where $A$ is diagonalisable, as is the case here. The restriction of a diagonalisable linear operator to any invariant subspace is always diagonalisable, which implies that such a subspace is equal to the direct sum of its intersections with the eigenspaces of the operator. So the subspace is then precisely determined by specifying a subspace of each eigenspace. If in addition, as is the case here, all eigenspaces are of dimension$~1$, there are only two subspaces in each case: $\{0\}$ and the whole eigenspace. For $3$ such eigenspaces one therefore gets $2^3=8$ invariant subspaces.
In general there may well be infinitely many invariant subspaces, as happens for instance whenever an eigenspace has dimension more than$~1$. The most delicate case for classifying invariant subspaces is that of nilpotent operators (or those obtained by adding a scalar matrix to them).
