Matrix I - 2 b b^T/(b^T b) where vector b is a any vector, has the following properties:

Note 1: a matrix is symmetric if it is equal to its transpose.

Note 2: a matrix is skew symmetric if it is equal to the negative of its transpose. Or equivalently, the sum of the matrix and its transpose is the zero matrix.

Note 3: a matrix is orthogonal if it is equal to the inverse (inv) of its transpose. Or equivalently, the product of the matrix and its transpose is the identity (eye) matrix.

Note 4: a matrix is idempotent if it is equal to the the square of itself (i.e. A = A²)

Note 8: Check all properties using the norm trick.

a.

Skew symmetric, orthogonal and has determinant equal to -1 (minus one)

b.

Skew symmetric, idempotent and has determinant equal to zero

c.

Symmetric, orthogonal and has determinant equal to 1 (one)

d.

Symmetric, orthogonal and has determinant equal to zero

e.

Symmetric, idempotent and has determinant equal to -1 (minus one)

f.

Symmetric, orthogonal and has determinant equal to -1 (minus one)

g.

Skew symmetric, orthogonal and has determinant equal to 1 (one)

h.

Symmetric, idempotent and has determinant equal to 1 (one)