The correct statements are:
1. For any positive integer k and any vector 𝐱∈ℝ𝑛 , we have 𝐻^𝑘𝐱=𝐻𝐱.
This is true because 𝐻 is an orthogonal projection, so raising it to any power does not change its action on vectors.
3. The matrix 𝐻 is a projection onto the subspace of vectors perpendicular to the vector 𝟏∈ℝ𝑛 , which has all of its entries equal to 1.
This is true because 𝐻 subtracts the average of the vector from each component, effectively making the vector perpendicular to the constant vector 𝟏 .
4. The matrix 𝐻 is a projection onto the subspace {𝐱:1𝑛∑𝑖=1𝑛𝐱𝑖=0}⊂ℝ𝑛 .
(In other words, this is the set of vectors having coordinate-wise average equal to 0.)
This is true because the matrix 𝐻 subtracts the average of the vector from each component, effectively ensuring that the coordinate-wise average of the resulting vector is zero.
Let \mathrm{{\boldsymbol X}}_1, \ldots , \mathrm{{\boldsymbol X}}_ n \in \mathbb {R}^ d denote a data set and let
\mathbb {X} = \begin{pmatrix} \longleftarrow & \mathbf{X}_1^ T & \longrightarrow \\ \longleftarrow & \mathbf{X}_2^ T & \longrightarrow \\ \vdots & \vdots & \vdots \\ \longleftarrow & \mathbf{X}_ n^ T & \longrightarrow \\ \end{pmatrix}.
Recall that the empirical covariance matrix S of this data set can be expressed as
S = \frac{1}{n} \mathbb {X}^ T H \mathbb {X}
where
H = I_ n - \frac{1}{n} \mathbf{1} \mathbf{1}^ T.
The matrix H \in \mathbb {R}^{n \times n} is an orthogonal projection .
In general, we say that a matrix M is an orthogonal projection onto a subspace S if
M is symmetric,
M^2 = M, and
S = \{ \mathrm{{\boldsymbol y}} : \, M \mathbf{x}= y \, \, \text {for some} \, \, \mathbf{x}\in \mathbb {R}^ n \}
Which of the following are true about the matrix H? (Choose all that apply.)
For any positive integer k and any vector \mathbf{x}\in \mathbb {R}^ n, we have H^ k \mathbf{x}= H \mathbf{x}.
For any positive integer k and any vector \mathbf{x}\in \mathbb {R}^ n, we have H^ k \mathbf{x}= \mathbf{x}.
The matrix H is a projection onto the subspace of vectors perpendicular to the vector \mathbf{1} \in \mathbb {R}^ n, which has all of its entries equal to 1.
The matrix H is a projections onto the subspace \{ \mathbf{x}: \frac{1}{n} \sum _{i = 1}^ n \mathbf{x}^ i = 0\} \subset \mathbb {R}^ n. (In other words, this is the set of vectors having coordinate-wise average equal to 0.)
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