linear algebra - Name for a matrix selecting one eigenvalue of an eigendecomposition

Let $\mathbf A$ be a square diagonalizable matrix and its eigendecomposition be given by
$$\mathbf A = \mathbf Q \mathbf \Lambda \mathbf Q^{-1} ,$$

where $\mathbf Q$ is a matrix of eigenvectors of $\mathbf A$, and $\mathbf \Lambda$ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues of $\mathbf A$. Let $\mathbf E_i$ be a matrix of the same size as $\mathbf A$ with zeros everywhere, except for the $(i,i)$th element being equal to $1$.

Does the matrix $\mathbf Q \mathbf E_i \mathbf Q^{-1}$ have a name?

Answer

The matrix $P= QE_iQ^{-1}$ is a projection matrix (since $P^2=P$). It plucks out the component of any vector along the $i$th eigenvector $Q_i$: if $v = \sum_j \alpha_j Q_j,$ then
$$Pv = \alpha_i Q_i.$$

When $Q$ is orthogonal, $P$ is simple Euclidean projection onto the vector $Q_i$. Otherwise, $P$ is projection onto $Q_i$ with respect to the inner product $Q^{-T}Q^{-1}$.