# Homography – Part II: useful facts

In order to prove the second largest singular value of Homography matrix is 1, or decompose the Homography matrix, the following facts are very important.

**Fact 1: **Let and . A vector makes if and only if or .

**Proof: **

That is

Hence implies .

People usually confuse Fact 1 and Fact 2. Just note for a symmetric matrix, we can not have implies . Compare the Fact 1 and Fact 2.

**Fact 2**: Let and . A vector makes if and only if and .

**Proof: **Sufficiency is apparent. Next prove necessity:

**case 1**: and are collinear. Then , hence . In general, . So it is easy to show has two zeros eigenvalues , and associating eivenvectors are perpendicular to .

**case 2**: and are not collinear.

**a) **if , then , . Easy

**b)** There exists but no perpendicular to . Then

**c)** There exists but no perpendicular to . Then

So if , then . Hence the eigenspace associating zero eigenvalue is one dimensional. So implies is collinear with .

In summary, if and only if and .

**Fact 3:** The homography matrix is . Let is a vector. Then if and only if .

**Proof**: is equivalent to . That is

where . According to the Fact 1, we have .