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:
“No being exists or can exist which is not related to space in some way. God is everywhere, created minds are somewhere, and body is in the space that it occupies; and whatever is neither everywhere nor anywhere does not exist. And hence it follows that space is an effect arising from the first existence of being, because when any being is postulated, space is postulated.”
—Isaac Newton (1642–1727)
“As for types like my own, obscurely motivated by the conviction that our existence was worthless if we didn’t make a turning point of it, we were assigned to the humanities, to poetry, philosophy, painting—the nursery games of humankind, which had to be left behind when the age of science began. The humanities would be called upon to choose a wallpaper for the crypt, as the end drew near.”
—Saul Bellow (b. 1915)
“Things perceived by the senses are immediately perceived by the senses; and things immediately perceived by the senses are ideas; and ideas cannot exist without the mind, their existence therefore consists in being perceived; when therefore they are actually perceived, there can be no doubt of their existence.”
—George Berkeley (1685–1753)