Multiplication of projection matrices is still a projection?

Question

Let $A,B$ be two projection matrices of the same dimension (i.e. in $\mathbb{R}^{n \times n}$), is their multiplication also a projection? (Is $AB$ a projection?)

Let $U,V \in \mathbb{R}^{d \times m}$ be two none square orthogonal matrices then where $m \leq d$. An additional question would be, if we assume that $A = U U^T$ and $B = V V^T$, its clear that $A$ and $B$ are matrices representing orthogonal projection, yet the question is, would $AB$ would represent an orthogonal projection?

please advise.

Answer 1

The product of two orthogonal matrices is orthogonal. If $A$ and $B$ are orthogonal, then

$$(AB)^T(AB)=B^TA^TAB=B^TB=I$$

The product of two projection matrices THAT COMMUTE is a projection. If $A$ and $B$ are projections and commute, then

$$(AB)^2=ABAB=AABB=A^2B^2=AB$$

In general, the product of two projection matrices need not be a projection.

Answer 2

Let consider

$$\left[\begin{array}{cc}1 &0\\0 &0\end{array}\right]\left[\begin{array}{cc}\frac12&\frac12\\\frac12 &\frac12\end{array}\right]=\left[\begin{array}{cc}\frac12 &\frac12\\0 &0\end{array}\right]$$