Existence
The question of whether a matrix has a logarithm has the easiest answer when considered in the complex setting. A matrix has a logarithm if and only if it is invertible. The logarithm is not unique, but if a matrix has no negative real eigenvalues, then it has a unique logarithm whose eigenvalues lie all in the strip {z ∈ C | −π < Im z < π}. This logarithm is known as the principal logarithm.
The answer is more involved in the real setting. A real matrix has a real logarithm if and only if it is invertible and each Jordan block belonging to a negative eigenvalue occurs an even number of times. If an invertible real matrix does not satisfy the condition with the Jordan blocks, then it has only complex logarithms. This can already be seen in the scalar case: the logarithm of −1 is a complex number. The existence of real matrix logarithms of real 2 x 2 matrices is considered in a later section.
Read more about this topic: Logarithm Of A Matrix
Famous quotes containing the word existence:
“It is because everything is relative
That we shall never see in that sphere of pure wisdom and
Entertainment much more than groping shadows of an incomplete
Former existence so close it burns like the mouth that
Closes down over all your effort like the moment
Of death”
—John Ashbery (b. 1927)
“I create my social existence by earning and spending.”
—Mason Cooley (b. 1927)
“Sometimes I think that the biggest difference between men and women is that more men need to seek out some terrible lurking thing in existence and hurl themselves upon it.... Women know where it lives but they can let it alone.”
—Russell Hoban (b. 1925)