Embeddings, Projections, and Inverses

I just revised a post from a week ago about rotations. The revision makes explicit the process of embedding a 3D vector into the quaternions, then pulling it back out.

The 3D vector is embedded in the quaternions by making it the vector part of a quaternion with zero real part:

(p₁,p₂,p₃) (0, p₁,p₂,p₃)

and the quaternion is returned to 3D by cutting off the real part:

(p₀, p₁,p₂,p₃) (p₁,p₂,p₃).

To give names to the the process of moving to and from quaterions, we have an embeddingE : ³ to ⁴ and a projectionP from ⁴ to ³.

We can represent E as a 4 * 3 matrix

and P by a 3 * 4 matrix

We'd like to sayE andP are inverses. SurelyP undoes whatE does, so they're inverses in that sense. ButE cannot undoP because you lose information projecting from ⁴ to ³ andE cannot recover information that was lost.

The rest of this post will look at three generalizations of inverses and howE andP relate to each.

Left and right inverse

Neither matrix is invertible, but PE equals the identity matrix on ³ , and soP is aleft inverse ofE andE is aright inverse ofP.

On the other hand,EP is not an identity matrix, and soE isnot a left inverse ofE, and neither isP a right inverse ofE.

Adjoint

P is the transpose ofE, which means it is the adjoint ofE. Adjoints are another way to generalize the idea of an inverse. More on that here.

Pseudo-inverse

The Moore-Penrose pseudo-inverse acts a lot like an inverse, which is somewhat uncanny because all matrices have a pseudo-inverse, even rectangular matrices.

Pseudo-inverses are symmetric, i.e. if A⁺ is the pseudo-inverse ofA, thenA is the pseudo-inverse of A⁺.

Given anmbynmatrixA, the Moore-Penrose pseudoinverseA⁺is the uniquenbymmatrix satisfying four conditions:

1. AA⁺A=A
2. A⁺AA⁺=A⁺
3. (AA⁺)* =AA⁺
4. (A⁺A)* =A⁺A

To show that A⁺ = P we have to establish

1. EPE=E
2. PEP=A⁺
3. (EP)* = EP
4. (PE)* = PE

We calculatedEP andPE above, and both are real and symmetric, so properties 3 and 4 hold.

We can also compute

and

showing that properties 1 and 2 hold as well.

Related posts

• Pseudo-inverse and SVD
• Random projections
• Applied linear algebra

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