Statement of Liouville's Formula
Consider the n-dimensional first-order homogeneous linear differential equation
on an interval I of the real line, where A(x) for x ∈ I denotes a square matrix of dimension n with real or complex entries. Let Φ denote a matrix-valued solution on I, meaning that each Φ(x) is a square matrix of dimension n with real or complex entries and the derivative satisfies
Let
denote the trace of A(ξ) = (ai, j (ξ))i, j ∈ {1,...,n}, the sum of its diagonal entries. If the trace of A is a continuous function, then the determinant of Φ satisfies
for all x and x0 in I.
Read more about this topic: Liouville's Formula
Famous quotes containing the words statement of, statement and/or formula:
“I think, therefore I am is the statement of an intellectual who underrates toothaches.”
—Milan Kundera (b. 1929)
“The honor my country shall never be stained by an apology from me for the statement of truth and the performance of duty; nor can I give any explanation of my official acts except such as is due to integrity and justice and consistent with the principles on which our institutions have been framed.”
—Andrew Jackson (1767–1845)
““It’s hard enough to adjust [to the lack of control] in the beginning,” says a corporate vice president and single mother. “But then you realize that everything keeps changing, so you never regain control. I was just learning to take care of the belly-button stump, when it fell off. I had just learned to make formula really efficiently, when Sarah stopped using it.””
—Anne C. Weisberg (20th century)