Transcription of Diagonal Matrices, Upper and Lower Triangular Matrices
1 Diagonal Matrices , Upper and Lower Triangular MatricesLinear AlgebraMATH 2010 Diagonal Matrices : Definition:Adiagonal matrixis a square matrix with zero entries except possibly on the maindiagonal (extends from the Upper left corner to the Lower right corner). Examples:The following are examples, of Diagonal Matrices : 1 0 00 1 00 0 1 120 0 00 3 0 00 0 0 00 0 0 4 In general, a Diagonal matrix is given byA= d10..00d20..00..00..00.. dk Notation:A lot of the time, a Diagonal matrix is referenced with a capitalD(for Diagonal ). Powers:IfDis a Diagonal matrix, thenDnforn >0 is given byDn= dn10..00dn20..00..00..00.. dnk Inverses:A Diagonal matrixDis invertible if and only if all the Diagonal elements are this case,D 1is given byD 1= 1d10..001d20..00..00..00..1dk Example:LetD= 120 0 00 3 0 00 0 5 00 0 0 1 ThenD3= (12)30 000 33000 0 5300 0 0 ( 1)3 = 180 0 00 27 0 00 0 125 00 0 0 1 andD 1= 2 0 0 00130 00 01500 0 0 1 Upper and Lower Triangular Matrices : Definition:Anupper Triangular matrixis a square matrix in which all entries below the maindiagonal are zero (only nonzero entries are found above the main Diagonal - in the Upper triangle).
2 Alower Triangular matrixis a square matrix in which all entries above the main Diagonal are zero(only nonzero entries are found below the main Diagonal - in the Lower triangle). See the picturebelow. Notation:An Upper Triangular matrix is typically denoted withUand a Lower Triangular matrixis typically denoted withL. Properties:1.{UT=LLT=UIf you transpose an Upper ( Lower ) Triangular matrix, you get a Lower ( Upper ) Triangular {L1L2=LU1U2=UThe product of two Lower ( Upper ) Triangular Matrices if Lower ( Upper ) A Triangular matrix is invertible if and only if all Diagonal entries are nonzero. 1 5 3 40 2 1 00 0 0 50 0 0 1 is NOT invertible, and 4 0 01 3 00 2 1 IS invertible.}}
