Transcription of Linear Algebra - CoAS
1 Linear AlgebraGrinshpanPermutation matricesA permutation matrix is a square matrix obtained from the same size identity matrix by apermutation of rows. Such a matrix is always row equivalent to an row and every column of a permutation matrix contain exactly one nonzero entry,which is are two 2 2 permutation matrices: [1 00 1],[0 11 0].There are six 3 3 permutation matrices. There aren! permutation matrices of permutation matrix is a product of elementary row-interchange matrices. The ele-mentary matrix factors may be chosen to only involve adjacent rows. For instance, 0 1 00 0 11 0 0 = 1 0 00 0 1 0 1 0 0 1 01 0 00 0 1 .Since interchangingith andjth rows of an identity is equivalent to interchanging itsithandjth columns, everyelementarypermutation matrix is symmetric,P>= general permutation matrix is not interchanging two rows is a self-reverse operation, everyelementarypermutation matrixis invertible and agrees with its inverse,P=P 1orP2= general permutation matrix does not agree with its product of permutation matrices is again a permutation inverse of a permutation matrix is again a permutation matrix .
2 In fact,P 1=P>.Left multiplication by a permutation matrix rearranges the corresponding rows: 0 1 00 0 11 0 0 x1x2x3 = x2x3x1 , 0 1 00 0 11 0 0 a a ab b bc c c = b b bc c ca a a .Right multiplication by a permutation matrix rearranges the corresponding columns: a b ca b ca b c 0 1 00 0 11 0 0 = c a bc a bc a b , 0 1 00 0 11 0 0 a b cd e fg h i 0 1 00 0 11 0 0 = f d ei g hc a b .Some power of a permutation matrix is the identity. For instance, 0 1 00 0 11 0 0 3= 1 0 00 1 00 0 1 .HereP3=IorP2=P 1=P>.
