#install.packages("matlib")
#install.packages("Ryacas")
#install.packages("MASS")
#install.packages("knitr")
#install.packages("xtable")
library(MASS)
library(matlib)
library(Ryacas)
##
## Adjuntando el paquete: 'Ryacas'
## The following objects are masked from 'package:matlib':
##
## tr, vec
## The following object is masked from 'package:stats':
##
## integrate
## The following objects are masked from 'package:base':
##
## %*%, det, diag, diag<-, lower.tri, upper.tri
#primer punto desarrollado con el uso de matrix
#a
m1<-matrix(c(1,21,11,-12,-13,11,41,23,-11,14,13,2,-14,-25,13,23,18,21,-17,-15)
,nrow = 4
,byrow = TRUE)
m1
## [,1] [,2] [,3] [,4] [,5]
## [1,] 1 21 11 -12 -13
## [2,] 11 41 23 -11 14
## [3,] 13 2 -14 -25 13
## [4,] 23 18 21 -17 -15
range(m1)
## [1] -25 41
#b
m2<-matrix(c(12,24,-23,18,10,20,10,-11,15,11,24,-31)
,nrow = 4
,byrow = TRUE)
m2
## [,1] [,2] [,3]
## [1,] 12 24 -23
## [2,] 18 10 20
## [3,] 10 -11 15
## [4,] 11 24 -31
range(m2)
## [1] -31 24
#c
m3<-matrix(c(12,23,10,-11,27,-31,-24,36)
,nrow = 4
,byrow = TRUE)
m3
## [,1] [,2]
## [1,] 12 23
## [2,] 10 -11
## [3,] 27 -31
## [4,] -24 36
range(m3)
## [1] -31 36
#d
m4<-matrix(c(15,17,12,18,-20,13,21,10,14,-15,18,17)
,nrow = 4
,byrow = TRUE)
m4
## [,1] [,2] [,3]
## [1,] 15 17 12
## [2,] 18 -20 13
## [3,] 21 10 14
## [4,] -15 18 17
range(m4)
## [1] -20 21
#desarrollo del segundo punto
a<-matrix(c(41,22,-15,16,12,27,37,19,22)
,nrow = 3
,ncol = 3
,byrow = TRUE)
a
## [,1] [,2] [,3]
## [1,] 41 22 -15
## [2,] 16 12 27
## [3,] 37 19 22
b<-matrix(c(12,13,-32,31,-17,32,21,-11,20)
,nrow = 3
,ncol = 3
,byrow = TRUE)
b
## [,1] [,2] [,3]
## [1,] 12 13 -32
## [2,] 31 -17 32
## [3,] 21 -11 20
c<-matrix(c(12,23,-18,21,10,21,11,-21,17)
,nrow = 3
,ncol = 3
,byrow = TRUE)
c
## [,1] [,2] [,3]
## [1,] 12 23 -18
## [2,] 21 10 21
## [3,] 11 -21 17
#traspuestas de las matrices
t(a)
## [,1] [,2] [,3]
## [1,] 41 16 37
## [2,] 22 12 19
## [3,] -15 27 22
t(b)
## [,1] [,2] [,3]
## [1,] 12 31 21
## [2,] 13 -17 -11
## [3,] -32 32 20
t(c)
## [,1] [,2] [,3]
## [1,] 12 21 11
## [2,] 23 10 -21
## [3,] -18 21 17
#inversa de las matrices
solve(a)
## [,1] [,2] [,3]
## [1,] -0.04065306 -0.125551020 0.12636735
## [2,] 0.10563265 0.237877551 -0.21991837
## [3,] -0.02285714 0.005714286 0.02285714
solve(b)
## [,1] [,2] [,3]
## [1,] 0.03896104 0.2987013 -0.4155844
## [2,] 0.16883117 2.9610390 -4.4675325
## [3,] 0.05194805 1.3149351 -1.9707792
solve(c)
## [,1] [,2] [,3]
## [1,] 0.042572464 -0.0009057971 0.04619565
## [2,] -0.008779264 0.0280100334 -0.04389632
## [3,] -0.038391862 0.0351867336 -0.02529264
#por propiedad la multiplicacion de la inversa por su matriz original debe ser la matriz identidad
#(los valores de las matrices si corresponden a la matriz identidad solo que estos se encuentran en notacion cientifica)
solve(a)%*%a
## [,1] [,2] [,3]
## [1,] 1.000000e+00 4.440892e-16 -8.881784e-16
## [2,] 0.000000e+00 1.000000e+00 1.776357e-15
## [3,] 1.110223e-16 1.110223e-16 1.000000e+00
solve(b)%*%b
## [,1] [,2] [,3]
## [1,] 1.000000e+00 3.552714e-15 -7.105427e-15
## [2,] -1.421085e-14 1.000000e+00 -1.421085e-14
## [3,] 0.000000e+00 0.000000e+00 1.000000e+00
solve(c)%*%c
## [,1] [,2] [,3]
## [1,] 1.000000e+00 2.220446e-16 0.000000e+00
## [2,] 0.000000e+00 1.000000e+00 1.110223e-16
## [3,] -1.665335e-16 -1.110223e-16 1.000000e+00
#adjunta de la matriz
adjoint(a)
## [,1] [,2] [,3]
## [1,] -249 -769 774
## [2,] 647 1457 -1347
## [3,] -140 35 140
adjoint(b)
## [,1] [,2] [,3]
## [1,] 12 92 -128
## [2,] 52 912 -1376
## [3,] 16 405 -607
adjoint(c)
## [,1] [,2] [,3]
## [1,] 611 -13 663
## [2,] -126 402 -630
## [3,] -551 505 -363
#podemos utilizar la propiedad donde la inversa de la matriz es igual a la adjunta divida el determinante
solve(a)
## [,1] [,2] [,3]
## [1,] -0.04065306 -0.125551020 0.12636735
## [2,] 0.10563265 0.237877551 -0.21991837
## [3,] -0.02285714 0.005714286 0.02285714
1/det(a)*adjoint(a)
## [,1] [,2] [,3]
## [1,] -0.04065306 -0.125551020 0.12636735
## [2,] 0.10563265 0.237877551 -0.21991837
## [3,] -0.02285714 0.005714286 0.02285714
solve(b)
## [,1] [,2] [,3]
## [1,] 0.03896104 0.2987013 -0.4155844
## [2,] 0.16883117 2.9610390 -4.4675325
## [3,] 0.05194805 1.3149351 -1.9707792
1/det(b)*adjoint(b)
## [,1] [,2] [,3]
## [1,] 0.03896104 0.2987013 -0.4155844
## [2,] 0.16883117 2.9610390 -4.4675325
## [3,] 0.05194805 1.3149351 -1.9707792
solve(c)
## [,1] [,2] [,3]
## [1,] 0.042572464 -0.0009057971 0.04619565
## [2,] -0.008779264 0.0280100334 -0.04389632
## [3,] -0.038391862 0.0351867336 -0.02529264
1/det(c)*adjoint(c)
## [,1] [,2] [,3]
## [1,] 0.042572464 -0.0009057971 0.04619565
## [2,] -0.008779264 0.0280100334 -0.04389632
## [3,] -0.038391862 0.0351867336 -0.02529264
#resolvemos los punto posteriores
#1
det(a%*%b)
## [1] 1886500
det(a)*det(b)
## [1] 1886500
#2
det(a)
## [1] 6125
det(t(a))
## [1] 6125
#3
det(b%*%c)
## [1] 4420416
det(b)*det(c)
## [1] 4420416
#4
det(b)
## [1] 308
det(t(b))
## [1] 308
#5
#establecemos una matriz identidad para los ejersicios posteriores
i<-diag(1,3,3)
i
## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 0 1 0
## [3,] 0 0 1
adjoint(a)%*%a
## [,1] [,2] [,3]
## [1,] 6125 0 0
## [2,] 0 6125 0
## [3,] 0 0 6125
det(a)*i
## [,1] [,2] [,3]
## [1,] 6125 0 0
## [2,] 0 6125 0
## [3,] 0 0 6125
#6
adjoint(c)%*%c
## [,1] [,2] [,3]
## [1,] 14352 0 0
## [2,] 0 14352 0
## [3,] 0 0 14352
det(c)*i
## [,1] [,2] [,3]
## [1,] 14352 0 0
## [2,] 0 14352 0
## [3,] 0 0 14352
#desarrollo del tercer punto
#a
mo<-matrix(c(15,3,2,-7),nrow = 2,ncol = 2, byrow = TRUE)
mo
## [,1] [,2]
## [1,] 15 3
## [2,] 2 -7
det(mo)
## [1] -111
#X y Y son iguales al determinante que se obtiene reemplazándolos por los terminos independientes
#dividido el determinante de la ecuación original
msx<-matrix(c(5,3,1,-7),nrow = 2,ncol = 2, byrow = TRUE)
msx
## [,1] [,2]
## [1,] 5 3
## [2,] 1 -7
det(msx)
## [1] -38
det(msx)/det(mo)
## [1] 0.3423423
msy<-matrix(c(15,5,2,1),nrow = 2,ncol = 2, byrow = TRUE)
msy
## [,1] [,2]
## [1,] 15 5
## [2,] 2 1
det(msy)
## [1] 5
det(msy)/det(mo)
## [1] -0.04504505
#b
mo2<-matrix(c(4,-1,7,2),nrow = 2,ncol = 2, byrow = TRUE)
mo2
## [,1] [,2]
## [1,] 4 -1
## [2,] 7 2
det(mo2)
## [1] 15
msx2<-matrix(c(10,-1,3,2),nrow = 2,ncol = 2, byrow = TRUE)
msx2
## [,1] [,2]
## [1,] 10 -1
## [2,] 3 2
det(msx2)
## [1] 23
det(msx2)/det(mo2)
## [1] 1.533333
msy2<-matrix(c(4,10,7,3),nrow = 2,ncol = 2, byrow = TRUE)
msy2
## [,1] [,2]
## [1,] 4 10
## [2,] 7 3
det(msy2)
## [1] -58
det(msy2)/det(mo2)
## [1] -3.866667
#c
#en este caso buscamos a y b para despues hallar X y Y
mab<-matrix(c(1,-2,3,-5),nrow = 2,ncol = 2, byrow = TRUE)
mab
## [,1] [,2]
## [1,] 1 -2
## [2,] 3 -5
det(mab)
## [1] 1
msa<-matrix(c(1,-2,2,-5),nrow = 2,ncol = 2, byrow = TRUE)
msa
## [,1] [,2]
## [1,] 1 -2
## [2,] 2 -5
a<-det(msa)/det(mab)
a
## [1] -1
msb<-matrix(c(1,1,3,2),nrow = 2,ncol = 2, byrow = TRUE)
msb
## [,1] [,2]
## [1,] 1 1
## [2,] 3 2
det(msb)
## [1] -1
b<-det(msb)/det(mab)
b
## [1] -1
#en este caso el sistema de ecuaciones tiene infinitas porque tiene mas incognitas
#que ecuaciones y ademas a y b son iguales en el caso de que c=1
#d
mo3<-matrix(c(3,5,4,-2),nrow = 2,ncol = 2, byrow = TRUE)
mo3
## [,1] [,2]
## [1,] 3 5
## [2,] 4 -2
det(mo3)
## [1] -26
msx3<-matrix(c(8,5,1,-2),nrow = 2,ncol = 2, byrow = TRUE)
msx3
## [,1] [,2]
## [1,] 8 5
## [2,] 1 -2
det(msx3)
## [1] -21
det(msx3)/det(mo3)
## [1] 0.8076923
msy3<-matrix(c(3,8,4,1),nrow = 2,ncol = 2, byrow = TRUE)
msy3
## [,1] [,2]
## [1,] 3 8
## [2,] 4 1
det(msy3)
## [1] -29
det(msy3)/det(mo3)
## [1] 1.115385
#e
mo4<-matrix(c(-7,-1,4,2),nrow = 2,ncol = 2, byrow = TRUE)
mo4
## [,1] [,2]
## [1,] -7 -1
## [2,] 4 2
det(mo4)
## [1] -10
msx4<-matrix(c(5,-1,-2,2),nrow = 2,ncol = 2, byrow = TRUE)
msx4
## [,1] [,2]
## [1,] 5 -1
## [2,] -2 2
det(msx4)
## [1] 8
det(msx4)/det(mo4)
## [1] -0.8
msy4<-matrix(c(-7,5,4,-2),nrow = 2,ncol = 2, byrow = TRUE)
msy4
## [,1] [,2]
## [1,] -7 5
## [2,] 4 -2
det(msy4)
## [1] -6
det(msy4)/det(mo4)
## [1] 0.6
#desarrollo del cuarto punto
#a
A<-matrix(c(13,21,33,14,-11,23,-12,-35,11),nrow = 3,ncol = 3, byrow = TRUE)
A
## [,1] [,2] [,3]
## [1,] 13 21 33
## [2,] 14 -11 23
## [3,] -12 -35 11
t(A)
## [,1] [,2] [,3]
## [1,] 13 14 -12
## [2,] 21 -11 -35
## [3,] 33 23 11
det(A)
## [1] -20664
solve(A)
## [,1] [,2] [,3]
## [1,] -0.03310105 0.067073171 -0.040940767
## [2,] 0.02080914 -0.026084011 -0.007888115
## [3,] 0.03010066 -0.009823848 0.021147890
adjoint(t(A))
## [,1] [,2] [,3]
## [1,] 684 -430 -622
## [2,] -1386 539 203
## [3,] 846 163 -437
#b
B<-matrix(c(12,23,10,14,-22,-13,27,13,21),nrow = 3,ncol = 3, byrow = TRUE)
B
## [,1] [,2] [,3]
## [1,] 12 23 10
## [2,] 14 -22 -13
## [3,] 27 13 21
t(B)
## [,1] [,2] [,3]
## [1,] 12 14 27
## [2,] 23 -22 13
## [3,] 10 -13 21
det(B)
## [1] -10591
solve(B)
## [,1] [,2] [,3]
## [1,] 0.02766500 0.033330186 0.007459163
## [2,] 0.06090076 0.001699556 -0.027948258
## [3,] -0.07326976 -0.043905203 0.055329997
adjoint(t(B))
## [,1] [,2] [,3]
## [1,] -293 -645 776
## [2,] -353 -18 465
## [3,] -79 296 -586
#c
C<-matrix(c(18,20,11,13,-12,-13,-21,-13,-17),nrow = 3,ncol = 3, byrow = TRUE)
C
## [,1] [,2] [,3]
## [1,] 18 20 11
## [2,] 13 -12 -13
## [3,] -21 -13 -17
t(C)
## [,1] [,2] [,3]
## [1,] 18 13 -21
## [2,] 20 -12 -13
## [3,] 11 -13 -17
det(C)
## [1] 5879
solve(C)
## [,1] [,2] [,3]
## [1,] 0.005953393 0.03350910 -0.02177241
## [2,] 0.084027896 -0.01275727 0.06412655
## [3,] -0.071610818 -0.03163803 -0.08096615
adjoint(t(C))
## [,1] [,2] [,3]
## [1,] 35 494 -421
## [2,] 197 -75 -186
## [3,] -128 377 -476
#d
D<-matrix(c(21,10,20,19,-21,-34,14,25,21),nrow = 3,ncol = 3, byrow = TRUE)
D
## [,1] [,2] [,3]
## [1,] 21 10 20
## [2,] 19 -21 -34
## [3,] 14 25 21
t(D)
## [,1] [,2] [,3]
## [1,] 21 19 14
## [2,] 10 -21 25
## [3,] 20 -34 21
det(D)
## [1] 15219
solve(D)
## [,1] [,2] [,3]
## [1,] 0.02687430 0.01905513 0.005256587
## [2,] -0.05749392 0.01057888 0.071883829
## [3,] 0.05052894 -0.02529733 -0.041461331
adjoint(t(D))
## [,1] [,2] [,3]
## [1,] 409 -875 769
## [2,] 290 161 -385
## [3,] 80 1094 -631
#e
E<-matrix(c(-12,10,-21,21,10,17,14,22,17),nrow = 3,ncol = 3, byrow = TRUE)
E
## [,1] [,2] [,3]
## [1,] -12 10 -21
## [2,] 21 10 17
## [3,] 14 22 17
t(E)
## [,1] [,2] [,3]
## [1,] -12 21 14
## [2,] 10 10 22
## [3,] -21 17 17
det(E)
## [1] -5504
solve(E)
## [,1] [,2] [,3]
## [1,] 0.03706395 0.11482558 -0.06904070
## [2,] 0.02162064 -0.01635174 0.04305959
## [3,] -0.05850291 -0.07340116 0.05995640
adjoint(t(E))
## [,1] [,2] [,3]
## [1,] -204 -119 322
## [2,] -632 90 404
## [3,] 380 -237 -330
#f
ys <- ysym
a <- ys("a")
b <- ys("b")
c <- ys("c")
col1<-c(a,c,b)
col2<-c(b,a,c)
col3<-c(c,b,a)
f<-cbind(col1,col2,col3)
f
## {{a, b, c},
## {c, a, b},
## {b, c, a}}
t(f)
## {{a, c, b},
## {b, a, c},
## {c, b, a}}
solve(f)
## {{1/a+(c*b)/(a^2*(a-(b*c)/a))+((((c-b^2/a)*c)/(a*(a-(b*c)/a))-b/a)*((b*(b-c^2/a))/((a-(b*c)/a)*a)-c/a))/(a-(c*b)/a-((b-c^2/a)*(c-b^2/a))/(a-(b*c)/a)), (-(((b*(b-c^2/a))/((a-(b*c)/a)*a)-c/a)*(c-b^2/a))/(a-(b*c)/a))/(a-(c*b)/a-((b-c^2/a)*(c-b^2/a))/(a-(b*c)/a))-b/(a*(a-(b*c)/a)), ((b*(b-c^2/a))/((a-(b*c)/a)*a)-c/a)/(a-(c*b)/a-((b-c^2/a)*(c-b^2/a))/(a-(b*c)/a))},
## { (-((((c-b^2/a)*c)/(a*(a-(b*c)/a))-b/a)*(b-c^2/a))/(a-(b*c)/a))/(a-(c*b)/a-((b-c^2/a)*(c-b^2/a))/(a-(b*c)/a))-c/(a*(a-(b*c)/a)), 1/(a-(b*c)/a)+((c-b^2/a)*(b-c^2/a))/((a-(b*c)/a)^2*(a-(c*b)/a-((b-c^2/a)*(c-b^2/a))/(a-(b*c)/a))), (-(b-c^2/a)/(a-(b*c)/a))/(a-(c*b)/a-((b-c^2/a)*(c-b^2/a))/(a-(b*c)/a))},
## { (((c-b^2/a)*c)/(a*(a-(b*c)/a))-b/a)/(a-(c*b)/a-((b-c^2/a)*(c-b^2/a))/(a-(b*c)/a)), (-(c-b^2/a)/(a-(b*c)/a))/(a-(c*b)/a-((b-c^2/a)*(c-b^2/a))/(a-(b*c)/a)), 1/(a-(c*b)/a-((b-c^2/a)*(c-b^2/a))/(a-(b*c)/a))}}
det(f)
## y: a^3-a*b*c+c^3-b*c*a+b^3-c*a*b
adj_f<-det(f)%*%solve(f)
t(adj_f)
## {{a^3-a*b*c+c^3-b*c*a+b^3-c*a*b*(1/a+(c*b)/(a^2*(a-(b*c)/a))+((((c-b^2/a)*c)/(a*(a-(b*c)/a))-b/a)*((b*(b-c^2/a))/((a-(b*c)/a)*a)-c/a))/(a-(c*b)/a-((b-c^2/a)*(c-b^2/a))/(a-(b*c)/a))), a^3-a*b*c+c^3-b*c*a+b^3-c*a*b*((-((((c-b^2/a)*c)/(a*(a-(b*c)/a))-b/a)*(b-c^2/a))/(a-(b*c)/a))/(a-(c*b)/a-((b-c^2/a)*(c-b^2/a))/(a-(b*c)/a))-c/(a*(a-(b*c)/a))), a^3-a*b*c+c^3-b*c*a+b^3-(c*a*b*(((c-b^2/a)*c)/(a*(a-(b*c)/a))-b/a))/(a-(c*b)/a-((b-c^2/a)*(c-b^2/a))/(a-(b*c)/a))},
## { a^3-a*b*c+c^3-b*c*a+b^3-c*a*b*((-(((b*(b-c^2/a))/((a-(b*c)/a)*a)-c/a)*(c-b^2/a))/(a-(b*c)/a))/(a-(c*b)/a-((b-c^2/a)*(c-b^2/a))/(a-(b*c)/a))-b/(a*(a-(b*c)/a))), a^3-a*b*c+c^3-b*c*a+b^3-c*a*b*(1/(a-(b*c)/a)+((c-b^2/a)*(b-c^2/a))/((a-(b*c)/a)^2*(a-(c*b)/a-((b-c^2/a)*(c-b^2/a))/(a-(b*c)/a)))), a^3-a*b*c+c^3-b*c*a+b^3-(-(c*a*b*(c-b^2/a))/(a-(b*c)/a))/(a-(c*b)/a-((b-c^2/a)*(c-b^2/a))/(a-(b*c)/a))},
## { a^3-a*b*c+c^3-b*c*a+b^3-(c*a*b*((b*(b-c^2/a))/((a-(b*c)/a)*a)-c/a))/(a-(c*b)/a-((b-c^2/a)*(c-b^2/a))/(a-(b*c)/a)), a^3-a*b*c+c^3-b*c*a+b^3-(-(c*a*b*(b-c^2/a))/(a-(b*c)/a))/(a-(c*b)/a-((b-c^2/a)*(c-b^2/a))/(a-(b*c)/a)), a^3-a*b*c+c^3-b*c*a+b^3-(c*a*b)/(a-(c*b)/a-((b-c^2/a)*(c-b^2/a))/(a-(b*c)/a))}}
#g
G<-matrix(c(13,17,-14,-23,10,-22,17,13,13),nrow = 3,ncol = 3,byrow = TRUE)
G
## [,1] [,2] [,3]
## [1,] 13 17 -14
## [2,] -23 10 -22
## [3,] 17 13 13
t(G)
## [,1] [,2] [,3]
## [1,] 13 -23 17
## [2,] 17 10 13
## [3,] -14 -22 13
det(G)
## [1] 10699
solve(G)
## [,1] [,2] [,3]
## [1,] 0.038882139 -0.03766707 -0.02187120
## [2,] -0.007010001 0.03804094 0.05682774
## [3,] -0.043835873 0.01121600 0.04869614
adjoint(t(G))
## [,1] [,2] [,3]
## [1,] 416 -75 -469
## [2,] -403 407 120
## [3,] -234 608 521
#h
H<-matrix(c(21,15,-17,-22,-12,-26,18,25,11,23,-11,22,-21,-16,28,13),
nrow = 4,ncol = 4,byrow = TRUE)
H
## [,1] [,2] [,3] [,4]
## [1,] 21 15 -17 -22
## [2,] -12 -26 18 25
## [3,] 11 23 -11 22
## [4,] -21 -16 28 13
t(H)
## [,1] [,2] [,3] [,4]
## [1,] 21 -12 11 -21
## [2,] 15 -26 23 -16
## [3,] -17 18 -11 28
## [4,] -22 25 22 13
det(H)
## [1] 188874
solve(H)
## [,1] [,2] [,3] [,4]
## [1,] 0.1043870517 0.05619619 0.01982274 0.035039233
## [2,] -0.0007571185 -0.03890424 0.02349185 0.033779133
## [3,] 0.0825576840 0.01142561 0.01882207 0.085887946
## [4,] -0.0101231509 0.01828732 0.02039455 -0.009890191
adjoint(t(H))
## [,1] [,2] [,3] [,4]
## [1,] 19716 -143 15593 -1912
## [2,] 10614 -7348 2158 3454
## [3,] 3744 4437 3555 3852
## [4,] 6618 6380 16222 -1868
#i
I<-matrix(c(13,-27,-15,24,-15,22,17,-65,-12,24,10,-23,22,-13,-15,31),
nrow = 4,ncol = 4,byrow = TRUE)
I
## [,1] [,2] [,3] [,4]
## [1,] 13 -27 -15 24
## [2,] -15 22 17 -65
## [3,] -12 24 10 -23
## [4,] 22 -13 -15 31
t(I)
## [,1] [,2] [,3] [,4]
## [1,] 13 -15 -12 22
## [2,] -27 22 24 -13
## [3,] -15 17 10 -15
## [4,] 24 -65 -23 31
det(I)
## [1] -61635
solve(I)
## [,1] [,2] [,3] [,4]
## [1,] -0.10732538 0.034071550 -0.11413969 0.069846678
## [2,] 0.01374219 -0.009653606 0.08233958 0.030210108
## [3,] -0.23618074 0.007706660 -0.27581731 -0.005629918
## [4,] -0.03235175 -0.024499067 -0.01792813 -0.007365945
adjoint(t(I))
## [,1] [,2] [,3] [,4]
## [1,] 6615 -847 14557 1994
## [2,] -2100 595 -475 1510
## [3,] 7035 -5075 17000 1105
## [4,] -4305 -1862 347 454
#j
J<-matrix(c(11,22,17,22,16,14,31,15,18),nrow = 3,ncol = 3,byrow = TRUE)
J
## [,1] [,2] [,3]
## [1,] 11 22 17
## [2,] 22 16 14
## [3,] 31 15 18
t(J)
## [,1] [,2] [,3]
## [1,] 11 22 31
## [2,] 22 16 15
## [3,] 17 14 18
det(J)
## [1] -1128
solve(J)
## [,1] [,2] [,3]
## [1,] -0.06914894 0.1250000 -0.03191489
## [2,] -0.03368794 0.2916667 -0.19503546
## [3,] 0.14716312 -0.4583333 0.27304965
adjoint(t(J))
## [,1] [,2] [,3]
## [1,] 78 38 -166
## [2,] -141 -329 517
## [3,] 36 220 -308
#desarrollo del quinto punto
#a1
a1<-matrix(c(-15,22,11,32,-12,22,11,12,-35),nrow = 3,ncol = 3,byrow = TRUE)
a1
## [,1] [,2] [,3]
## [1,] -15 22 11
## [2,] 32 -12 22
## [3,] 11 12 -35
ai1<-Ginv(a1)
# Verificación de propiedades
av1 <- all.equal(a1 %*% ai1 %*% a1, a1)
bv1 <- all.equal(ai1 %*% a1 %*% ai1, ai1)
c_symv1 <- all.equal(a1 %*% ai1, t(a1 %*% ai1))
c_idempv1 <- all.equal((a1 %*% ai1)^2, a1 %*% ai1)
d_symv1 <- all.equal(ai1 %*% a1, t(ai1 %*% a1))
d_idempv1 <- all.equal((ai1 %*% a1)^2, ai1 %*% a1)
# verificaciones
#nota: para simetria e idempotencia se utiliza la definicion para verificar,donde
#simetria es a=a^t e idempotencia a=a^2
list(
ai1 = ai1,
Verificaciones = list(
"AA^+A = A" = av1,
"A^+AA^+ = A^+" = bv1,
"AA^+ simétrica" = c_symv1,
"AA^+ idempotente" = c_idempv1,
"A^+A simétrica" = d_symv1,
"A^+A idempotente" = d_idempv1
)
)
## $ai1
## [,1] [,2] [,3]
## [1,] 0.00468468 0.02708709 0.01849850
## [2,] 0.04090090 0.01213213 0.02048048
## [3,] 0.01549550 0.01267267 -0.01573574
##
## $Verificaciones
## $Verificaciones$`AA^+A = A`
## [1] "Mean relative difference: 2.023256e-07"
##
## $Verificaciones$`A^+AA^+ = A^+`
## [1] "Mean relative difference: 1.434456e-07"
##
## $Verificaciones$`AA^+ simétrica`
## [1] "Mean relative difference: 1.2"
##
## $Verificaciones$`AA^+ idempotente`
## [1] "Mean relative difference: 3.066666e-07"
##
## $Verificaciones$`A^+A simétrica`
## [1] "Mean relative difference: 1.259259"
##
## $Verificaciones$`A^+A idempotente`
## [1] "Mean relative difference: 4.299999e-07"
#a2
a2<-matrix(c(21,11,-11,31,10,-22),nrow = 2,ncol = 3,byrow = TRUE)
a2
## [,1] [,2] [,3]
## [1,] 21 11 -11
## [2,] 31 10 -22
ai2<-ginv(a2)
# Verificación de propiedades
av2 <- all.equal(a2 %*% ai2 %*% a2, a2)
bv2 <- all.equal(ai2 %*% a2 %*% ai2, ai2)
c_symv2 <- all.equal(a2 %*% ai2, t(a2 %*% ai2))
c_idempv2 <- all.equal((a2 %*% ai2)^2, a2 %*% ai2)
d_symv2 <- all.equal(ai2 %*% a2, t(ai2 %*% a2))
d_idempv2 <- all.equal((ai2 %*% a2)^2, ai2 %*% a2)
# Resultados
list(
ai2 = ai2,
Verificaciones = list(
"AA^+A = A" = av2,
"A^+AA^+ = A^+" = bv2,
"AA^+ simétrica" = c_symv2,
"AA^+ idempotente" = c_idempv2,
"A^+A simétrica" = d_symv2,
"A^+A idempotente" = d_idempv2
)
)
## $ai2
## [,1] [,2]
## [1,] 0.02746516 0.002234591
## [2,] 0.14149027 -0.085381709
## [3,] 0.10301467 -0.081115671
##
## $Verificaciones
## $Verificaciones$`AA^+A = A`
## [1] TRUE
##
## $Verificaciones$`A^+AA^+ = A^+`
## [1] TRUE
##
## $Verificaciones$`AA^+ simétrica`
## [1] TRUE
##
## $Verificaciones$`AA^+ idempotente`
## [1] TRUE
##
## $Verificaciones$`A^+A simétrica`
## [1] TRUE
##
## $Verificaciones$`A^+A idempotente`
## [1] "Mean relative difference: 1.244537"
#a3
a3<-matrix(c(11,20,13,12,-31,16),nrow = 2,ncol = 3,byrow = TRUE)
a3
## [,1] [,2] [,3]
## [1,] 11 20 13
## [2,] 12 -31 16
(ai3<-Ginv(a3))
## [,1] [,2]
## [1,] 0.00000000 0.00000000
## [2,] 0.02213001 -0.01798063
## [3,] 0.04287690 0.02766252
# Verificación de propiedades
av3 <- all.equal(a3 %*% ai3 %*% a3, a3)
bv3 <- all.equal(ai3 %*% a3 %*% ai3, ai3)
c_symv3 <- all.equal(a3 %*% ai3, t(a3 %*% ai3))
c_idempv3 <- all.equal((a3 %*% ai3)^2, a3 %*% ai3)
d_symv3 <- all.equal(ai3 %*% a3, t(ai3 %*% a3))
d_idempv3 <- all.equal((ai3 %*% a3)^2, ai3 %*% a3)
# Resultados
list(
ai3 = ai3,
Verificaciones = list(
"AA^+A = A" = av3,
"A^+AA^+ = A^+" = bv3,
"AA^+ simétrica" = c_symv3,
"AA^+ idempotente" = c_idempv3,
"A^+A simétrica" = d_symv3,
"A^+A idempotente" = d_idempv3
)
)
## $ai3
## [,1] [,2]
## [1,] 0.00000000 0.00000000
## [2,] 0.02213001 -0.01798063
## [3,] 0.04287690 0.02766252
##
## $Verificaciones
## $Verificaciones$`AA^+A = A`
## [1] "Mean relative difference: 1.701942e-07"
##
## $Verificaciones$`A^+AA^+ = A^+`
## [1] "Mean relative difference: 1.3175e-07"
##
## $Verificaciones$`AA^+ simétrica`
## [1] "Mean relative difference: 0.56"
##
## $Verificaciones$`AA^+ idempotente`
## [1] "Mean relative difference: 2.5e-07"
##
## $Verificaciones$`A^+A simétrica`
## [1] "Mean relative difference: 2"
##
## $Verificaciones$`A^+A idempotente`
## [1] "Mean relative difference: 0.06979973"
#a4
a4<-matrix(c(31,21,21,12,31,43),nrow = 3,ncol = 2,byrow = TRUE)
ai4<- ginv(a4)
# Verificación de propiedades
av4 <- all.equal(a4 %*% ai4 %*% a4, a)
bv4 <- all.equal(ai4 %*% a4 %*% ai4, ai4)
c_symv4 <- all.equal(a4 %*% ai4, t(a4 %*% ai4))
c_idempv4 <- all.equal((a4 %*% ai4)^2, a4 %*% ai4)
d_symv4 <- all.equal(ai4 %*% a4, t(ai4 %*% a4))
d_idempv4 <- all.equal((ai4 %*% a4)^2, ai4 %*% a4)
# Resultados
list(
ai4 = ai4,
Verificaciones = list(
"AA^+A = A" = av4,
"A^+AA^+ = A^+" = bv4,
"AA^+ simétrica" = c_symv4,
"AA^+ idempotente" = c_idempv4,
"A^+A simétrica" = d_symv4,
"A^+A idempotente" = d_idempv4
)
)
## $ai4
## [,1] [,2] [,3]
## [1,] 0.03790404 0.03229651 -0.02752425
## [2,] -0.02619286 -0.02473911 0.04295162
##
## $Verificaciones
## $Verificaciones$`AA^+A = A`
## [1] "Modes: numeric, list"
## [2] "Lengths: 6, 1"
## [3] "names for current but not for target"
## [4] "Attributes: < Names: 1 string mismatch >"
## [5] "Attributes: < Component 1: Modes: numeric, character >"
## [6] "Attributes: < Component 1: target is numeric, current is character >"
## [7] "target is matrix, current is yac_symbol"
##
## $Verificaciones$`A^+AA^+ = A^+`
## [1] TRUE
##
## $Verificaciones$`AA^+ simétrica`
## [1] TRUE
##
## $Verificaciones$`AA^+ idempotente`
## [1] "Mean relative difference: 0.6008275"
##
## $Verificaciones$`A^+A simétrica`
## [1] TRUE
##
## $Verificaciones$`A^+A idempotente`
## [1] TRUE
#a5
a5<-matrix(c(33,-11,24,21,-21,31,-15,-21,24,-25,22,10),nrow = 3,ncol = 4,
byrow = TRUE)
a5
## [,1] [,2] [,3] [,4]
## [1,] 33 -11 24 21
## [2,] -21 31 -15 -21
## [3,] 24 -25 22 10
(ai5<-ginv(a5))
## [,1] [,2] [,3]
## [1,] 0.021560191 0.01568767 0.007630095
## [2,] 0.035269347 0.02504892 -0.024144079
## [3,] 0.003987863 0.03452995 0.043480048
## [4,] 0.027655609 -0.05099398 -0.074328532
# Verificación de propiedades
av5 <- all.equal(a5 %*% ai5 %*% a5, a5)
bv5 <- all.equal(ai5 %*% a5 %*% ai5, ai5)
c_symv5 <- all.equal(a5 %*% ai5, t(a5 %*% ai5))
c_idempv5 <- all.equal((a5 %*% ai5)^2, a5 %*% ai5)
d_symv5 <- all.equal(ai5 %*% a5, t(ai5 %*% a5))
d_idempv5 <- all.equal((ai5 %*% a5)^2, ai5 %*% a5)
# Resultados
list(
ai5 = ai5,
Verificaciones = list(
"AA^+A = A" = av5,
"A^+AA^+ = A^+" = bv5,
"AA^+ simétrica" = c_symv5,
"AA^+ idempotente" = c_idempv5,
"A^+A simétrica" = d_symv5,
"A^+A idempotente" = d_idempv5
)
)
## $ai5
## [,1] [,2] [,3]
## [1,] 0.021560191 0.01568767 0.007630095
## [2,] 0.035269347 0.02504892 -0.024144079
## [3,] 0.003987863 0.03452995 0.043480048
## [4,] 0.027655609 -0.05099398 -0.074328532
##
## $Verificaciones
## $Verificaciones$`AA^+A = A`
## [1] TRUE
##
## $Verificaciones$`A^+AA^+ = A^+`
## [1] TRUE
##
## $Verificaciones$`AA^+ simétrica`
## [1] TRUE
##
## $Verificaciones$`AA^+ idempotente`
## [1] TRUE
##
## $Verificaciones$`A^+A simétrica`
## [1] TRUE
##
## $Verificaciones$`A^+A idempotente`
## [1] "Mean relative difference: 0.7306335"
#a6
a6<-matrix(c(21,22,34,43,13,-21,32,-22,15,-24,10,-27),nrow = 3,ncol = 4,
byrow = TRUE)
a6
## [,1] [,2] [,3] [,4]
## [1,] 21 22 34 43
## [2,] 13 -21 32 -22
## [3,] 15 -24 10 -27
(ai6<-ginv(a6))
## [,1] [,2] [,3]
## [1,] 0.0234593646 -0.047420253 0.071009724
## [2,] 0.0011268116 0.003161384 -0.017137539
## [3,] -0.0006967407 0.043663229 -0.037514858
## [4,] 0.0117733179 -0.012983137 0.003751786
# Verificación de propiedades
av6 <- all.equal(a6 %*% ai6 %*% a6, a6)
bv6 <- all.equal(ai6 %*% a6 %*% ai6, ai6)
c_symv6 <- all.equal(a6 %*% ai6, t(a6 %*% ai6))
c_idempv6 <- all.equal((a6 %*% ai6)^2, a6 %*% ai6)
d_symv6 <- all.equal(ai6 %*% a6, t(ai6 %*% a6))
d_idempv6 <- all.equal((ai6 %*% a6)^2, ai6 %*% a6)
# Resultados
list(
ai6 = ai6,
Verificaciones = list(
"AA^+A = A" = av6,
"A^+AA^+ = A^+" = bv6,
"AA^+ simétrica" = c_symv6,
"AA^+ idempotente" = c_idempv6,
"A^+A simétrica" = d_symv6,
"A^+A idempotente" = d_idempv6
)
)
## $ai6
## [,1] [,2] [,3]
## [1,] 0.0234593646 -0.047420253 0.071009724
## [2,] 0.0011268116 0.003161384 -0.017137539
## [3,] -0.0006967407 0.043663229 -0.037514858
## [4,] 0.0117733179 -0.012983137 0.003751786
##
## $Verificaciones
## $Verificaciones$`AA^+A = A`
## [1] TRUE
##
## $Verificaciones$`A^+AA^+ = A^+`
## [1] TRUE
##
## $Verificaciones$`AA^+ simétrica`
## [1] TRUE
##
## $Verificaciones$`AA^+ idempotente`
## [1] TRUE
##
## $Verificaciones$`A^+A simétrica`
## [1] TRUE
##
## $Verificaciones$`A^+A idempotente`
## [1] "Mean relative difference: 0.6058805"
#a7
a7<-matrix(c(-21,10,31,22,
-31,11,20,-41,
20,-11,31,23,
20,21,-41,-13,
31,-21,30,11,
21,10,-21,-22),nrow = 6,ncol = 4,byrow = TRUE)
a7
## [,1] [,2] [,3] [,4]
## [1,] -21 10 31 22
## [2,] -31 11 20 -41
## [3,] 20 -11 31 23
## [4,] 20 21 -41 -13
## [5,] 31 -21 30 11
## [6,] 21 10 -21 -22
(ai7<-ginv(a7))
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.001131089 -0.0003108506 0.010212388 0.0106853495 0.011292300
## [2,] 0.030303854 0.0066207549 0.011613980 0.0245308954 -0.005038925
## [3,] 0.010776442 0.0110700084 0.010055641 0.0006203432 0.008429429
## [4,] 0.010705712 -0.0155014731 0.002553451 0.0017203228 -0.006092993
## [,6]
## [1,] 0.011719069
## [2,] 0.013092071
## [3,] 0.004506835
## [4,] -0.007253258
# Verificación de propiedades
av7 <- all.equal(a7 %*% ai7 %*% a7, a7)
bv7 <- all.equal(ai7 %*% a7 %*% ai7, ai7)
c_symv7 <- all.equal(a7 %*% ai7, t(a7 %*% ai7))
c_idempv7 <- all.equal((a7 %*% ai7)^2, a7 %*% ai7)
d_symv7 <- all.equal(ai7 %*% a7, t(ai7 %*% a7))
d_idempv7 <- all.equal((ai7 %*% a7)^2, ai7 %*% a7)
# Resultados
list(
ai7 = ai7,
Verificaciones = list(
"AA^+A = A" = av7,
"A^+AA^+ = A^+" = bv7,
"AA^+ simétrica" = c_symv7,
"AA^+ idempotente" = c_idempv7,
"A^+A simétrica" = d_symv7,
"A^+A idempotente" = d_idempv7
)
)
## $ai7
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.001131089 -0.0003108506 0.010212388 0.0106853495 0.011292300
## [2,] 0.030303854 0.0066207549 0.011613980 0.0245308954 -0.005038925
## [3,] 0.010776442 0.0110700084 0.010055641 0.0006203432 0.008429429
## [4,] 0.010705712 -0.0155014731 0.002553451 0.0017203228 -0.006092993
## [,6]
## [1,] 0.011719069
## [2,] 0.013092071
## [3,] 0.004506835
## [4,] -0.007253258
##
## $Verificaciones
## $Verificaciones$`AA^+A = A`
## [1] TRUE
##
## $Verificaciones$`A^+AA^+ = A^+`
## [1] TRUE
##
## $Verificaciones$`AA^+ simétrica`
## [1] TRUE
##
## $Verificaciones$`AA^+ idempotente`
## [1] "Mean relative difference: 1.232263"
##
## $Verificaciones$`A^+A simétrica`
## [1] TRUE
##
## $Verificaciones$`A^+A idempotente`
## [1] TRUE
#a8
a8<-matrix(c(-16,22,-32,-13,23,-31,15,22,-13,21,33,-33),nrow = 3,ncol = 4,
byrow = TRUE)
a8
## [,1] [,2] [,3] [,4]
## [1,] -16 22 -32 -13
## [2,] 23 -31 15 22
## [3,] -13 21 33 -33
(ai8<-ginv(a8))
## [,1] [,2] [,3]
## [1,] 0.09131997 0.10694295 0.03922986
## [2,] -0.05778614 -0.07416814 -0.02299418
## [3,] -0.06192659 -0.04847858 -0.00774403
## [4,] -0.13467412 -0.13780553 -0.06813391
# Verificación de propiedades
av8 <- all.equal(a8 %*% ai8 %*% a8, a8)
bv8 <- all.equal(ai8 %*% a8 %*% ai8, ai8)
c_symv8 <- all.equal(a8 %*% ai8, t(a8 %*% ai8))
c_idempv8 <- all.equal((a8 %*% ai8)^2, a8 %*% ai8)
d_symv8 <- all.equal(ai8 %*% a8, t(ai8 %*% a8))
d_idempv8 <- all.equal((ai8 %*% a8)^2, ai8 %*% a8)
# Resultados
list(
ai8 = ai8,
Verificaciones = list(
"AA^+A = A" = av8,
"A^+AA^+ = A^+" = bv8,
"AA^+ simétrica" = c_symv8,
"AA^+ idempotente" = c_idempv8,
"A^+A simétrica" = d_symv8,
"A^+A idempotente" = d_idempv8
)
)
## $ai8
## [,1] [,2] [,3]
## [1,] 0.09131997 0.10694295 0.03922986
## [2,] -0.05778614 -0.07416814 -0.02299418
## [3,] -0.06192659 -0.04847858 -0.00774403
## [4,] -0.13467412 -0.13780553 -0.06813391
##
## $Verificaciones
## $Verificaciones$`AA^+A = A`
## [1] TRUE
##
## $Verificaciones$`A^+AA^+ = A^+`
## [1] TRUE
##
## $Verificaciones$`AA^+ simétrica`
## [1] TRUE
##
## $Verificaciones$`AA^+ idempotente`
## [1] TRUE
##
## $Verificaciones$`A^+A simétrica`
## [1] TRUE
##
## $Verificaciones$`A^+A idempotente`
## [1] "Mean relative difference: 0.8767003"
#desarrollo del sexto punto
#establecer las matrices
(ao<-matrix(c(15,22,14,14,-21,13),nrow = 2,ncol = 3,byrow = TRUE))
## [,1] [,2] [,3]
## [1,] 15 22 14
## [2,] 14 -21 13
#(bo<-matrix(c(1327,31," " ,1,2,-6),nrow = 2,ncol = 3,byrow = TRUE))
(co<-matrix(c(31,13,21,10,22,-13,20,30,15),nrow = 3,ncol = 3,byrow = TRUE))
## [,1] [,2] [,3]
## [1,] 31 13 21
## [2,] 10 22 -13
## [3,] 20 30 15
(do<-matrix(c(1,3,0,21,10,-24,10,21,13),nrow = 3,ncol = 3,byrow = TRUE))
## [,1] [,2] [,3]
## [1,] 1 3 0
## [2,] 21 10 -24
## [3,] 10 21 13
(eo<-matrix(c(31,13,21,16,
10,20,-13,7,
10,20,15,23,
14,18,22,21),nrow = 4,ncol = 4,byrow = TRUE))
## [,1] [,2] [,3] [,4]
## [1,] 31 13 21 16
## [2,] 10 20 -13 7
## [3,] 10 20 15 23
## [4,] 14 18 22 21
(fo<-matrix(c(11,30,20,11,20,-14,20,21,13),nrow = 3,ncol = 3,byrow = TRUE))
## [,1] [,2] [,3]
## [1,] 11 30 20
## [2,] 11 20 -14
## [3,] 20 21 13
(go<-matrix(c(21,13,10,22),nrow = 2,ncol = 2,byrow = TRUE))
## [,1] [,2]
## [1,] 21 13
## [2,] 10 22
(ho<-matrix(c(10,10,30,12),nrow = 2,ncol = 2,byrow = TRUE))
## [,1] [,2]
## [1,] 10 10
## [2,] 30 12
(io<-matrix(c(11,41,21,31),nrow = 2,ncol = 2,byrow = TRUE))
## [,1] [,2]
## [1,] 11 41
## [2,] 21 31
#realizar las operaciones
#a
(t(ao)%*%ao)
## [,1] [,2] [,3]
## [1,] 421 36 392
## [2,] 36 925 35
## [3,] 392 35 365
#b
#(t(bo)%*%bo)
#c
#(bo%*%t(bo))
#d
(t(co)%*%co)
## [,1] [,2] [,3]
## [1,] 1461 1223 821
## [2,] 1223 1553 437
## [3,] 821 437 835
#e
#(t(ao)+t(bo))
#f
(t((3*do)+(2*fo))+t(co))
## [,1] [,2] [,3]
## [1,] 56 95 90
## [2,] 82 92 135
## [3,] 61 -113 80
#g
(2*(eo^3)+3*(eo^2))
## [,1] [,2] [,3] [,4]
## [1,] 62465 4901 19845 8960
## [2,] 2300 17200 -3887 833
## [3,] 2300 17200 7425 25921
## [4,] 6076 12636 22748 19845
#h
(4*(go^4)+3*(ho))
## [,1] [,2]
## [1,] 777954 114274
## [2,] 40090 937060
#i
((ho^10)-50*(go))
## [,1] [,2]
## [1,] 9.999999e+09 9999999350
## [2,] 5.904900e+14 61917363124
#j
#(co%*%do)
(do%*%co)
## [,1] [,2] [,3]
## [1,] 61 79 -18
## [2,] 271 -227 -49
## [3,] 780 982 132
(fo%*%do)
## [,1] [,2] [,3]
## [1,] 841 753 -460
## [2,] 291 -61 -662
## [3,] 591 543 -335
(do%*%fo)
## [,1] [,2] [,3]
## [1,] 44 90 -22
## [2,] -139 326 -32
## [3,] 601 993 75
(ao%*%co)
## [,1] [,2] [,3]
## [1,] 965 1099 239
## [2,] 484 110 762
#(bo%*%co)
#k
(det(fo)*solve(fo)%*%t(fo))
## [,1] [,2] [,3]
## [1,] -9406 18174 1050
## [2,] -4883 -15029 -8995
## [3,] 7011 7061 2939
#l
(t(solve(eo))%*%solve(t(eo)))
## [,1] [,2] [,3] [,4]
## [1,] -0.001683372 0.01195604 0.01217276 -0.01906905
## [2,] 0.004375682 -0.01056330 -0.01070762 0.01430471
## [3,] -0.020055824 0.07807279 0.06266882 -0.10558972
## [4,] 0.018768072 -0.07965531 -0.06647695 0.11164163
#las operaciones que incluyen la matriz B son imposibles de realizar debido a la
#falta de un valor numerico que complete la matriz
#desarrollo del septimo punto
#nota: los valores propios se pueden hallar al despejar la formula
#det(A - λI) = 0 para lambda que seria los valores propios y para los
#vectores propios se usa (A - λI)v = 0 dendo se busca v
#establecemos una funcion que halle tanto los valores como los vectores propios
#mediante la funcion eigen (funcion que genera la descomposicion espectral de una matriz)
valp_vecp<- function(M) {
e <- eigen(M)
list(
valores_propios = e$values,
vectores_propios = e$vectors
)
}
#a
A7<-matrix(c(11,41,21,31),nrow=2,ncol=2,byrow = TRUE)
A7
## [,1] [,2]
## [1,] 11 41
## [2,] 21 31
valp_vecp(A7)
## $valores_propios
## [1] 52 -10
##
## $vectores_propios
## [,1] [,2]
## [1,] -0.7071068 -0.8900434
## [2,] -0.7071068 0.4558759
#b
B7<-matrix(c(-23,10,-21,-17,15,-21,-16,26,-12),nrow=3,ncol=3,byrow = TRUE)
B7
## [,1] [,2] [,3]
## [1,] -23 10 -21
## [2,] -17 15 -21
## [3,] -16 26 -12
valp_vecp(B7)
## $valores_propios
## [1] -20.4303436+ 0.00000i 0.2151718+11.82141i 0.2151718-11.82141i
##
## $vectores_propios
## [,1] [,2] [,3]
## [1,] -0.8626289+0i -0.33660708+0.4054648i -0.33660708-0.4054648i
## [2,] -0.4894952+0i 0.09650507+0.5433765i 0.09650507-0.5433765i
## [3,] -0.1275377+0i 0.64631469+0.0000000i 0.64631469+0.0000000i
#c
C7<-matrix(c(21,-21,12,-21),nrow=2,ncol=2,byrow = TRUE)
C7
## [,1] [,2]
## [1,] 21 -21
## [2,] 12 -21
valp_vecp(C7)
## $valores_propios
## [1] 13.74773 -13.74773
##
## $vectores_propios
## [,1] [,2]
## [1,] 0.9452218 0.5172344
## [2,] 0.3264289 0.8558438
#d
D7<-matrix(c(15,-21,31,13),nrow=2,ncol=2,byrow = TRUE)
D7
## [,1] [,2]
## [1,] 15 -21
## [2,] 31 13
valp_vecp(D7)
## $valores_propios
## [1] 14+25.4951i 14-25.4951i
##
## $vectores_propios
## [,1] [,2]
## [1,] 0.02490677+0.6350006i 0.02490677-0.6350006i
## [2,] 0.77211000+0.0000000i 0.77211000+0.0000000i
#e
E7<-matrix(c(13,-21,13,-23),nrow=2,ncol=2,byrow = TRUE)
E7
## [,1] [,2]
## [1,] 13 -21
## [2,] 13 -23
valp_vecp(E7)
## $valores_propios
## [1] -12.141428 2.141428
##
## $vectores_propios
## [,1] [,2]
## [1,] 0.6410633 0.8882780
## [2,] 0.7674880 0.4593062
#f
F7<-matrix(c(13,21,31,12,24,12,21,11,31),nrow=3,ncol=3,byrow = TRUE)
F7
## [,1] [,2] [,3]
## [1,] 13 21 31
## [2,] 12 24 12
## [3,] 21 11 31
valp_vecp(F7)
## $valores_propios
## [1] 59.076106 15.556897 -6.633002
##
## $vectores_propios
## [,1] [,2] [,3]
## [1,] 0.6306581 0.01999518 0.8807580
## [2,] 0.4355098 -0.82698661 -0.1722118
## [3,] 0.6423408 0.56186595 -0.4411444
#g
G7<-matrix(c(21,31,20,31),nrow=2,ncol=2,byrow = TRUE)
G7
## [,1] [,2]
## [1,] 21 31
## [2,] 20 31
valp_vecp(G7)
## $valores_propios
## [1] 51.3968502 0.6031498
##
## $vectores_propios
## [,1] [,2]
## [1,] -0.7140188 -0.8353909
## [2,] -0.7001265 0.5496562
#h
H7<-matrix(c(1, 1i,0, 1i), nrow = 2, byrow = TRUE)
H7
## [,1] [,2]
## [1,] 1+0i 0+1i
## [2,] 0+0i 0+1i
valp_vecp(H7)
## $valores_propios
## [1] 1+0i 0+1i
##
## $vectores_propios
## [,1] [,2]
## [1,] 1+0i 0.4082483-0.4082483i
## [2,] 0+0i 0.8164966+0.0000000i
#i
I7<-matrix(c(20,10,11,10,21,20,11,20,8),nrow=3,ncol=3,byrow = TRUE)
I7
## [,1] [,2] [,3]
## [1,] 20 10 11
## [2,] 10 21 20
## [3,] 11 20 8
valp_vecp(I7)
## $valores_propios
## [1] 44.309919 11.635699 -6.945618
##
## $vectores_propios
## [,1] [,2] [,3]
## [1,] 0.5162737 0.8455259 -0.1361891
## [2,] 0.6743983 -0.4993862 -0.5438753
## [3,] 0.5278716 -0.1889428 0.8280412
#j
J7<-matrix(c(12,-13,17,-14),nrow=2,ncol=2,byrow = TRUE)
J7
## [,1] [,2]
## [1,] 12 -13
## [2,] 17 -14
valp_vecp(J7)
## $valores_propios
## [1] -1+7.211103i -1-7.211103i
##
## $vectores_propios
## [,1] [,2]
## [1,] 0.5756497+0.319313i 0.5756497-0.319313i
## [2,] 0.7527727+0.000000i 0.7527727+0.000000i