We conclude that every rotation matrix, when expressed in a suitable coordinate system, partitions into independent rotations of two-dimensional subspaces, at most of them. The sum of the entries on the main diagonal of a matrix is called the trace; it does not change if we reorient the coordinate system, and always equals the sum of the eigenvalues. This has the convenient implication for and rotation matrices that the trace reveals the angle of rotation, , in the two-dimensional space (or subspace). For a matrix the trace is , and for a matrix it is . In the three-dimensional case, the subspace consists of all vectors perpendicular to the rotation axis (the invariant direction, with eigenvalue 1). Thus we can extract from any rotation matrix a rotation axis and an angle, and these completely determine the rotation. Sequential angles The constraints on a rotation matrix imply that it must have the form with . Therefore, we may set and , for some angle . To solve for it is not enough to look at alone or alone; we must consider both together to place the angle in the correct quadrant, using a two-argument arctangent function. Now consider the first column of a rotation matrix, Although will probably not equal 1, but some value , we can use a slight variation of the previous computation to find a so-called Givens rotation that transforms the column to zeroing . This acts on the subspace spanned by the - and -axes. We can then repeat the process for the -subspace to zero . Acting on the full matrix, these two rotations produce the schematic form Shifting attention to the second column, a Givens rotation of the -subspace can now zero the value. This brings the full matrix to the form which is an identity matrix. Thus we have decomposed as An rotation matrix will have , or entries below the diagonal to zero. We can zero them by extending the same idea of stepping through the columns with a series of rotations in a fixed sequence of planes. We conclude that the set of rotation matrices, each of which has entries, can be parameterized by angles. In three dimensions this restates in matrix form an observation made by Euler, so mathematicians call the ordered sequence of three angles Euler angles. However, the situation is somewhat more complicated than we have so far indicated.