QS-014 · Linear Algebra · Easy

Properties of a Projection Matrix onto a 2D Subspace

Let $M\in\mathbb{R}^{3\times3}$ and let $U$ be a $2$-dimensional subspace such that $M$ is a projection onto $U$. Which of the following statements are true? 1. $M^3=M$ 2. $M^2=M$ 3. The nullspace of $M$ is 1-dimensional. 4. The nullspace of $M$ is 2-dimensional.

Let $M\in\mathbb{R}^{3\times3}$ and let $U$ be a $2$-dimensional subspace such that $M$ is a projection onto $U$. Which of the following statements are true? 1. $M^3=M$ 2. $M^2=M$ 3. The nullspace of $M$ is 1-dimensional. 4. The nullspace of $M$ is 2-dimensional.
Solution $M$ is a projection matrix. One of the properties of a projection matrix is that it is idempotent. So, $$ M^2=M $$ Multiplying both sides by $M$, $$ M^3=M^2=M. $$ Therefore, **options A and B are true**. Now, since $M$ is projecting onto a subspace with dimension 2, $$ \operatorname{rank}(M)=2. $$ By the Rank-Nullity Theorem, $$ \operatorname{rank}(M)+\operatorname{nullity}(M)=3. $$ Therefore, $$ \operatorname{nullity}(M)=3-2=1. $$ Hence, **option C is also true**, while option D is false. Therefore, **options A, B, and C are correct**.

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