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:
“Eroticism has its own moral justification because it says that pleasure is enough for me; it is a statement of the individual’s sovereignty.”
—Mario Vargas Llosa (b. 1936)
“Eloquence must be grounded on the plainest narrative. Afterwards, it may warm itself until it exhales symbols of every kind and color, speaks only through the most poetic forms; but first and last, it must still be at bottom a biblical statement of fact.”
—Ralph Waldo Emerson (1803–1882)
“I feel like a white granular mass of amorphous crystals—my formula appears to be isomeric with Spasmotoxin. My aurochloride precipitates into beautiful prismatic needles. My Platinochloride develops octohedron crystals,—with a fine blue florescence. My physiological action is not indifferent. One millionth of a grain injected under the skin of a frog produced instantaneous death accompanied by an orange blossom odor.”
—Lafcadio Hearn (1850–1904)