PROPERTIES OF THE L MATRIX
Since va and vb are physically the same vector v, the magnitude of va must be the same as that of vb, i. e. v2 is an invariant of the transformation. From (4.4,3) this requires
= УьТУь = v/LbaTLbava = vjva (4.4,6)
It follows from the last equality of (4.4,6) that
Equation (4.4,6) is known as the orthogonality condition on L6a. From (4.4,6) it follows that,
lLJ2 = 1
and hence that |Lba| is never zero and the inverse of Lba always exists. In view of (4.4,6) we have, of course, that
haT = 4a-1 = 4b (4-4,7)
i. e. that the inverse and the transpose are the same. Equation (4.4,6) together with (4.4,36) yields a set of conditions on the direction cosines, i. e.
2 kks = Ь» (4-4,8)
It follows from (4.4,8) that the columns of Lbaare vectors that form an orthogonal set (hence the name “orthogonal matrix”) and that they are of unit length.
Since (4.4,8) are a set of six relations among the nine lijy then only three of them are independent. These three are an alternative to the three independent Euler angles for specifying the orientation of one frame relative to another.