Andre Louis Cholesky

Andre Louis Cholesky was a prestigious member of the French army and a successful mathematician. He studied at the Ecole Polytechnique in France and remained in the upper half of his class upon graduating. Following his studies at the Ecole Polytechnique, Andre Louis Cholesky joined the French army and became a second lieutenant. While in the army, he studied at the school d'Application de l'Artillerie et du Génie where he was once again in the most intellectual part of his class. He spent some time serving his country in Tunisia where the French were able to stimulate the economy and establish modern communication. After serving a mission in Tunisia, he was transferred to Algeria where he served another mission. The French treated their efforts in Algeria completely differently than their efforts in Tunisia. They developed hospitals, medical services, and new communications, but dominantly took control of the country and its native people. Upon leaving Algeria, Cholesky joined the Geodesic section of the army geographic service. It was said that Cholesky had “ a sharp intelligence and a great facility for mathematical work, having an inquiring spirit and original ideas.” This field of the army put his true strengths of mathematical excellence to the test. His works had profound effects on the allocations of continents on the map. Cholesky produced a computational procedure, which was fairly simple compared to the previous works of Jean Baptiste Joseph Delombre. It can be said that Cholesky perfected his works. His procedure proved to be simple, fast, and precise and became known as the method of Cholesky. A few short months after his ingenious works, he became first lieutenant in the French army. Two years later, he married his wife who birthed his three children. His military efforts were profound in that he paved the way for the French army to build railroads in Tunisia, Algeria, and later Morocco. After serving 21 years in the army, Andre Louis Cholesky died from a fatal battlefield wound. His fellow officer Commandant Benoit published Cholesky's method of computing solutions in 1924, but it was not highly recognized until referenced in the works of Jack Todd during World War II. His statistical theories proved to be extremely useful. His computations were named the Cholesky Factorization or Cholesky Decomposition. The formula takes a symmetric positive definite matrix A and writes it as A = LL' where L is a lower triangular matrix with positive diagonal entries and L' is the transpose of L. To solve Ax = b one now needs to solve LL'x = b so put y = L'x which gives Ly = b which is solved for y, then y = L'x is solved for x to obtain the solution. Ultimately, his calculations are used to solve complex linear equations using triangular matrices. Andre Louis Cholesky was born on October 15th, 1875 in Montguyon (a city located northeast of Bordeaux, France). He is the son of Andre Cholesky and Marie Garnier, and he had several brothers and sisters growing up. Upon attending the Ecole Polytechnique, he signed a three year contract with the army. His family is said to have come to France with Napoleon’s army, so it was a given for him to combine his thirst for knowledge with his eagerness to serve a military and intellectual purpose. In 1909, Cholesky was appointed to captain and from this point on is when the manuscript on his method is dated. His methods were appreciated, and they helped to gain time in the leveling operations. In September of 1911, he was appointed to the artillery’s headquarters. Here, he took orders to conduct the leveling operations in Algeria and Tunisia, ultimately mapping the country and preparing it for railways and roads. In 1913, he became the head of topographical services in Tunis. With his consistent climb in rank Cholesky proved to be beneficial to the French army. He was soon put in charge of a battery where they were using maps at a scale of 1/80000. The measurements were far too broad and the soldiers were essentially firing at invisible targets. The target distances could not be accurately accounted for and changes obviously needed to be made. Cholesky used the advantage of aerial planes to take photographs of enemy lines where topographers could not reach. He learned how to develop precise maps by listening to the sounds of enemy batteries, aerial photography, and even the sights for shooting from an aeroplane. After noticing the weaknesses in their military tactics, Cholesky wrote many official documents on these problems, and even began to learn English. C. Berzinski, the man who wrote the first biography of Andre Louis Cholesky, defined the methods of triangulation, leveling, and compensation “The method of triangulation is used for establishing maps. It goes back to the 16th century at least. The region to be mapped is covered by triangles. For refinement, one can use networks of smaller and smaller triangles. One begins to measure the length of a side of the first triangle, (called the basis) and then, standing at each corner of the adjacent triangles, only angles are measured. It is easier and less subject to errors to measure angles than distances. Vertical angles have also to be measured for obtaining a map in an horizontal plane (an operation called levelling). Then, the usual trigonometric formulae (spherical trigonometry for long distances) give the lengths. Thus, the topographer obtains a network of adjacent triangles. Obviously, the three angles of each triangle (instead of two) could be measured for safety, and a second basis also, thus leading to more equations than unknowns. But one has to take into account the various sources of errors. This is called compensation. “ Cholesky used these tools to map the battlefields for the batteries of the Romanian and French armies. Using these methods allowed more precise attacks on incoming enemy batteries, and more precise orders to attack from aero planes.
Cholesky presented his method in an unknown and unpublished manuscript entitled Sur la r ́esolution num ́erique des syst`emes d’ ́equations lin ́eaires. It contains 8 pages, and is dated December 2nd, 1910, but is said to be a copy or rewrite of a preceding text. Berzinski explains the systems in Cholesky’s manuscript on the fifth page of his reflection http://math.univ-lille1.fr/~brezinsk/cholNUMA.pdf .
This is a statement from http://en.wikipedia.org/wiki/Cholesky_decomposition about Cholesky’s decomposition “The Cholesky decomposition of a Hermitian positive-definite matrix A is a decomposition of the form A=LL* where L is a lower triangular matrix with real and positive diagonal entries, and L* denotes the conjugate transpose of L. Every Hermitian positive-definite matrix (and thus also every real-valued symmetric positive-definite matrix) has a unique Cholesky decomposition.[2] If the matrix A is Hermitian and positive semi-definite, then it still has a decomposition of the form A = LL* if the diagonal entries of L are allowed to be zero.[3]
When A has real entries, L has real entries as well and the factorization may be written A = LLT [4] The Cholesky decomposition is unique when A is positive definite; there is only one lower triangular matrix L with strictly positive diagonal entries such that A = LL*. However, the decomposition need not be unique when A is positive semidefinite. The converse holds trivially: if A can be written as LL* for some invertible L, lower triangular or otherwise, then A is Hermitian and positive definite.
These are some examples of Cholesky’s complex formulas to solve linear equations. The equations can be used in many different instances, for example the targeting on a battlefield, but can also be used to map out the expansion of a railway system. Andre Louis Cholesky has had a profound effect on the expansion of power in France and a force in the battlefield with his strategic intelligence.

REFERENCES http://math.univ-lille1.fr/~brezinsk/cholNUMA.pdf http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Cholesky.html http://en.wikipedia.org/wiki/Cholesky_decomposition http://www.oxfordreference.com/view/10.1093/oi/authority.20110803095609582