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:
“How old the world is! I walk between two eternities.... What is my fleeting existence in comparison with that decaying rock, that valley digging its channel ever deeper, that forest that is tottering and those great masses above my head about to fall? I see the marble of tombs crumbling into dust; and yet I don’t want to die!”
—Denis Diderot (1713–1784)
“I consider, then, the power to annul a law of the United States, assumed by one state, incompatible with the existence of the Union, contradicted expressly by the letter of the Constitution, unauthorized by its spirit, inconsistent with every principle on which it was founded, and destructive of the great object for which it was formed.”
—Andrew Jackson (1767–1845)
“...care and labor are as much correlated to human existence as shadow is to light ...”
—Harriet Beecher Stowe (1811–1896)