projection and orthogonal matrices

for linear algebra

What is the length of the projection of vector v onto a unit vector u? Is it equal to their dot product?

The length of the projection is the absolute value of their dot product:
∣proj_u(v)∣ = ∣v⋅u∣.
The actual dot product v⋅u can be negative (if the angle is >90°), but length is always non-negative.

Key: Projection length requires absolute value; the dot product itself gives a signed scalar.

What does the matrix UU^T do when U has orthonormal columns?

• UU^T is the projection matrix onto the column space of U.

• For any vector y, UU^Ty projects y orthogonally onto the subspace spanned by U's columns.

• Key properties:

  • UU^T is idempotent (UU^T)²=UU^T.

  • UU^T is symmetric (UU^T)^T=(UU^T).

  • The residual y−UU^Ty is orthogonal to the subspace.

• Note: U^TU=I (since columns are orthonormal), but UUT≠I unless U is square (i.e., the subspace is the entire space).

Is vvT always a projection matrix for any nonzero vector v?

No, vv^T is a projection matrix only if v is a unit vector.
For general v, the correct projection matrix is vv^T/v^Tv.

Does the projection matrix onto a subspace depend on the basis chosen for that subspace?

No, the projection matrix onto a subspace is always the same, no matter which basis you use. It depends only on the subspace, not the basis.