Projector is a type of matrix For more detailed look into this mathematical entity, visit: [[Deeper Look into Projector]] --- * **Projector** * Standing Assumption: $P \in \mathbb{R}^{n\times n}$, $P^2 = P$ * Simply speaking: transforming vectors from the subspace spanned by the matrix onto itself is not changing the vectors in the subspace. * e.g: A single unit vector is a projector * Facts: * $P(Pv - v) = P^2v - Pv = 0$, and hence $(I - P)$ will be the complementary projector, because whatever it projected it's orthogonal to $Ps projection. * proof: $(I - P)$ is a projector because: * $(I - P)^2 = (I - 2P - P^2) = (I - P)$ * proof $(I - P)$ is complementary to $P$ because: * $(I - P)P = 0$ * Then $null(P)\subseteq range(I - P)\wedge range(I - P) \subseteq null(P)$ implying that: $range(I - P) = null(P)$ * **Ortho Projector**: * First of all, it's a prjector * It's orthogonal if: * $((I - P)y)^*(Px) = 0 \quad\forall x, y\in \mathbb{R^n}$, putting vector through the matrix will always results in orthogonal vectors. * Or simply put: $range(P) \perp range(I - P)$ * $P$ is an orthogonal prjector **iff** $P = P^*$ and $P^2 = P$ * --> Can be proved directly * <-- use [[SVD]] to show that P decompose to $Q\Sigma Q^*$ * [ ] Proof ...? --- #### **Example: Ortho Projector** Assume that the matrix $!$ is given as a unitary matrix, $Q \in \mathbb{C}^{m\times n}$ Ok, now if I want to represent any vector $x\in \mathbb{C}^{m}$ in the space, when we will do this: $ x = r + \sum_{j = 1}^n (q^H_jx)\circ q_j $ Where $\circ$ is the elementwise multiplications. Notice that this sum over there is a matrix multiplicaton in disguise: $QQ^Hx$ And yes, that is your projection matrix into the subspace, and the complementary operator will be presented as: $ I - QQ^H $ --- ### **Example: Penrose Pseudo Inverse** Penrose Pseudo Inverse is one projector matrix. Here is the definition of Penrose Paused Inverse matrix: >$ AA^\dagger = A(A^HA)^{-1}A$ Notice that it's not hard to check: > $AA^\dagger AA^\dagger =AA^\dagger$ And this is fitting the definition of a projection matrix.