MATRICES AND GAUSS-JORDAN Math 21b, O. KnillMATRIX FORMULATION. Consider the sys-tem of linear equations3x − y − z = 0−x + 2y − z = 0−x − y + 3z = 9The system can be written as A~x =~b, where A is a matrix(called coefficient matrix) and and ~x and~b are vectors.A =3 −1 −1−1 2 −1−1 −1 3, ~x =xyz,~b =009.((A~x)iis the dot product of the i’th row with ~x).We also look at the augmented matrixwhere one puts separators for clarity reasons.B =3 −1 −1 | 0−1 2 −1 | 0−1 −1 3 | 9.MATRIX ”JARGON”. A rectangular array of numbers is called amatrix. If the matrix has m rows and n columns, it is called am × n matrix. A matrix with one column only is called a columnvector, a matrix with one row a row vector. The entries of amatrix are denoted by aij, where i is the row and j is the column.In the case of the linear equation above, the matrix A is a squarematrix and the augmented matrix B above is a 3 × 4 matrix.mnGAUSS-JORDAN ELIMINATION. Gauss-Jordan Elimination is a process, where successive subtraction ofmultiples of other rows or scaling brings the matrix into reduced row echelon form. The elimination processconsists of three possible steps which are called elementary row operations:• Swap two rows.• Divide a row by a scalar• Subtract a multiple of a row from an other row.The process transfers a given matrix A into a new matrix rref(A)REDUCED ECHELON FORM. A matrix is called in reduced row echelon form1) if a row has nonzero entries, then the first nonzero entry is 1. (leading one)2) if a column contains a leading 1, then the other column entries are 0.3) if a row has a leading 1, then every row above has leading 1’s to the left.Pro memoriam: Leaders like to be number one, are lonely and want other leaders above to their left.RANK. The number of leading 1 in rref(A) is called the rank of A.SOLUTIONS OF LINEAR EQUATIONS. If Ax = b is a linear system of equations with m equations and nunknowns, then A is a m × n matrix. We have the following three possibilities:• Exactly one solution. There is a leading 1 in each row but not in the last row.• Zero solutions. There is a leading 1 in the last row.• Infinitely many solutions. There are rows without leading 1 and no leading 1 is in the last row.JIUZHANG SUANSHU. The technique of successivelyeliminating variables from systems of linear equationsis called Gauss elimination or Gauss Jordanelimination and appeared already in the Chinesemanuscript ”Jiuzhang Suanshu” (’Nine Chapters on theMathematical art’). The manuscript appeared around200 BC in the Han dynasty and was probably used asa textbook. For more history of Chinese Mathematics,seehttp://aleph0.clarku.edu/ ˜djoyce/mathhist/china.html.EXAMPLES. The reduced echelon form of the augmented matrixB determines on how many solutions the linear system Ax = bhas.THE GOOD (1 solution) THE BAD (0 solution) THE UGLY (∞ solutions)"0 1 2 | 21 −1 1 | 52 1 −1 | −2#"1 −1 1 | 50 1 2 | 22 1 −1 | −2#"1 −1 1 | 50 1 2 | 20 3 −3 | −12#"1 −1 1 | 50 1 2 | 20 1 −1 | −4#"1 0 3 | 70 1 2 | 20 0 −3 | −6#"1 0 3 | 70 1 2 | 20 0 1 | 2#1 0 0 | 101 0 | −20 01 | 2Rank(A) = 3, Rank(B) = 3.0 1 2 | 21 −1 1 | 51 0 3 | −21 −1 1 | 50 1 2 | 21 0 3 | −21 −1 1 | 50 1 2 | 20 1 2 | −71 −1 1 | 50 1 2 | 20 0 0 | −91 0 3 | 70 1 2 | 20 0 0 | −91 0 3 | 701 2 | 20 0 0 | 1Rank(A) = 2, Rank(B) = 3.0 1 2 | 21 −1 1 | 51 0 3 | 71 −1 1 | 50 1 2 | 21 0 3 | 71 −1 1 | 50 1 2 | 20 1 2 | 21 −1 1 | 50 1 2 | 20 0 0 | 01 0 3 | 701 2 | 20 0 0 | 0Rank(A) = 2, Rank(B) = 2.JORDAN. The German geodesist Wilhelm Jordan (1842-1899) applied the Gauss-Jordan method to findingsquared errors to work on surveying. (An other ”Jordan”, the French Mathematician Camille Jordan (1838-1922) worked on linear algebra topics also (Jordan form) and is often mistakenly credited with the Gauss-Jordanprocess.)GAUSS. Gauss developed Gaussian elimination around 1800 and used it tosolve least squares problems in celestial mechanics and later in geodesic compu-tations. In 1809, Gauss published the book ”Theory of Motion of the HeavenlyBodies” in which he used the method for solving astronomical problems. Oneof Gauss successes was the prediction of an asteroid orbit using linear algebra.CERES. On 1. January of 1801, the Italian astronomer Giuseppe Piazzi (1746-1826) discovered Ceres, the first and until 2001 the largest known asteroid in thesolar system. (A new found object called 2001 KX76 is estimated to have a 1200 kmdiameter, half the size of Pluto) Ceres is a rock of 914 km diameter. (The picturesCeres in infrared light). Gauss was able to predict the orbit of Ceres from a fewobservations. By parameterizing the orbit with parameters and solving a linearsystem of equations (similar to one of the homework problems, where you will fit acubic curve from 4 observations), he was able to derive the orbit
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