Semi-inverse Eigenvalues and Rotation Matrices

Donald R. Burleson, Ph.D.

It is well known in matrix theory that

is a rotation matrix having the effect, by matrix multiplication, of rotating a vector counterclockwise in the plane through an angle . As the theorem proven below will show, this concept of a rotation matrix interacts in an interesting way with the concept of the (principal) semi-inverse of a nonsingular matrix. In previous articles I have described the semi-inversion operator having the matrix property . I generally denote the principal semi-inverse of A as the matrix . I will also use the terminology that a complex number z “generates” the associated multiplication matrix

Let us now see how things interrelate by proving the following:

THEOREM: Given any semi-invertible matrix A and any eigenvalue of A, the corresponding eigenvalue of the (principal) semi-inverse of A generates a rotation matrix. Specifically, any complex-number eigenvalue z of A has a corresponding eigenvalue of that generates the stretching/contraction rotation matrix

PROOF: Given a semi-invertible matrix A, let us first consider the simple case in which an eigenvalue is a positive real number. Euler’s Formula implies that in principal complex value, the corresponding eigenvalue in the principal semi-inverse matrix is

and the associated multiplication matrix for this complex number is

which from its form is the rotation matrix that rotates a vector through an angle of radians.

The situation is only slightly different for a negative real eigenvalue in that again

but in principal absolute value the exponentiations, from Euler’s relation , are of the form

and this complex number generates the associated multiplication matrix

which is a contraction/rotation matrix that rotates a vector through an angle of radians and multiplies the length of the vector by

These two results, for a real-number eigenvalue, are of course special cases of the more general situation where an original A-eigenvalue is a not-necessarily-real complex number. We now examine that scenario to find, however, that again the results are similar.

For a complex A-eigenvalue, yet again Euler’s Formula in principal value implies that

, where now the logarithms are given by expressions of the form , and using the standard formulas for cos(A+B) and sin(A+B) along with the formulas (for complex numbers z)

it is a fairly straightforward computation to find, in principal value, that

and this complex number generates the matrix

which is a “stretching or contraction” (depending on the numerical size of the coefficient) rotation matrix that in the general case rotates a vector through an angle of radians, where the vertical bars this time indicate the modulus of the complex-number eigenvalue.

It is clear that the special cases give respectively the positive and negative real-number eigenvalue results previously shown.

This completes the proof, except that we have tacitly assumed, in the argument for negative real-number eigenvalues, that for two real numbers a and b,

But this is easily established, since the right-hand expression is given by

[cos ln(a) + isin ln(a)][cos ln(b) + isin ln(b)]

= cos ln(a) cos ln(b) + i[sin ln(a) cos ln(b) + sin ln(b) cos ln(a)] - sin ln(a) sin ln(b)

= cos [ ln(a) + ln(b) ] + isin[ln(a) + ln(b)] = cos ln(ab) + isin ln(ab)

which equals the left-hand expression above as needed.