Lecture Notes – Statistics 892 – Mixed Models – Spring 2005 – page 1IntroductionMatrix AlgebraDefinitionsA matrix is a rectangular array of numbers with n rows and m columns35343332312524232221151413121153aaaaaaaaaaaaaaaAxA vector is a column of number or an n rows by 1 column matrix654321aaaaaaa A scalar is a single number.A square matrix is a matrix with the number of rows (n) equal to the number of columns (m). In this case, the elements a11, a22, a33, a44,…, ann,are called the diagonal elements of the matrix and the sum of the diagonal elements is called the trace of the matrix.190005000333xA The trace of A=3+5+19=27.An identity matrix is a square matrix with all 1’s on the diagonal.10001000133xILecture Notes – Statistics 892 – Mixed Models – Spring 2005 – page 2A transpose of a matrix is the matrix whose columns are its rows and is indicated with a prime (`).538421a  538421a65194284333xA 69854412333xARows (or columns) are linear dependent if a row is a linear combination of one or more other rows in the matrix. 12511242144333xA 2*Column 1 + 2*Column 2=Column 3. The rank of any matrix is the number of linearly independent rows (or columns) of the matrix. Properties of rank1. Rank is always positive.2. For a rectangular matrix, the rank is always < the smaller of rows or columns.3. For a square matrix, the rank is always < its order.4. If the rank of a square matrix < its order, then its inverse does not exist. (More on inverses later)Matrix Addition and SubtractionThe sum of two matrices is the matrix of sums, element by elementLecture Notes – Statistics 892 – Mixed Models – Spring 2005 – page 310981099161254471578826519428433333BABAxx 216815041BAMatrices must have the same order in order to be conformable for addition or subtraction.Scalar MultiplicationGiven a scalar a=3 and the matrix A3x3 defined above181532712624129651942843aThus the matrix A multiplied by the scalar a is the matrix A with every element multiplied by a.Matrix MultiplicationLecture Notes – Statistics 892 – Mixed Models – Spring 2005 – page 424)23)(34(23341)1)(2()1)(3()2)(1(1)0)(2()0)(3()1)(1(3)1)(1()1)(2()2)(1(1)0)(1()0)(2()1)(1(6)1)(1()1)(1()2)(3(3)0)(1()0)(1()1)(3(0)1)(2()1)(0()2)(1(1)0)(2()0)(0()1)(1(101021231121113201xxxxxABBAThe jth column of B must have the same number of elements as the ith row of A in order for the matrices to be conformable for multiplication.Given A and c, we can solve for b with the following system of equationsAb = c => b=A-1cThe inverse of a matrix is the matrix whose product with the original matrix is the identity matrix. 65194284333xA 148148.04074074.0222222.04074074.037037.11111.0148148.592593.7777778.0331xAIf a matrix is full rank then its inverse exists. If the matrix is not full rank, it is called singular its inverse does not exist. Provided Ab=c has a solution, then a solution is b= A-cA generalized inverse of nxm matrix A, denoted by A-, is any mxn matrix that satisifies the following relationshipAA-A=ALecture Notes – Statistics 892 – Mixed Models – Spring 2005 – page 5Note that there are an infinite number of generalized inverses.Example:1001020200331236A 03020100b 27244596cb\s\up 7(0 ) b\s\up 7(0 ) b\s\up 7(0 ) b\s\up 7(0 )µ\s\up 7(0 ) 16 14 27 -2982\s\up 7(0 ) -1 1 -12 2997\s\up 7(0 ) -4 -2 -15 2994\s\up 7(0 ) 11 13 0 3009The four b0 presented above are four of an infinite number of solutions for the parameters. However, an investigator is normally not interested in the treatment solutions, but in specific expressions that can be described with a linear functions of the solutions of b0 including the following:\s\up 7(0 ) - \s\up 7(0 ) - difference of effects of two treatment levels = 3 for all four b0µ\s\up 7(0 ) + \s\up 7(0 ) - general mean plus effect of the first treatment level = 15½(\s\up 7(0 ) + \s\up 7(0 )) - \s\up 7(0 ) - superiority of mean effect of levels 2 and 3 over effect of level 1 = 5.5These linear functions are invariant to whatever solution is