Matrix Calculation in R

Add and subtract matrices> a = matrix(c(5, 2, 3, 1), nrow = 2, ncol = 2, byrow = T )> a

[, 1] [, 2][1, ] 5 2[2, ] 3 1> b = matrix(c(8, 2,5,7), 2, 2, byrow = T )> b

[, 1] [, 2][1, ] 8 2[2, ] 5 7> a+ b

[, 1] [, 2][1, ] 13 4[2, ] 2 6> a b

[, 1] [, 2][1, ] 3 0[2, ] 8 8

Multiplication by a scalar> c = matrix(c(1,2, 3, 0, 4,5), 2, 2, byrow = T )> c

[, 1] [, 2] [, 3][1, ] 1 2 3[2, ] 0 4 5> d = 2 c

> d[, 1] [, 2] [, 3]

[1, ] 2 4 6[2, ] 0 8 10

Transpose of a matrix> t(c)

[, 1] [, 2][1, ] 1 0[2, ] 2 4[3, ] 3 5

Matrix multiplication> a = matrix(c(3, 0,2, 1,1, 4), 2, 3, byrow = T )> a

[, 1] [, 2] [, 3][1, ] 3 0 2[2, ] 1 1 4> b = matrix(c(1, 1, 1, 2, 1, 3), 3, 2, byrow = T )> b

[, 1] [, 2][1, ] 1 1[2, ] 1 2[3, ] 1 3> c = a% %b> c

[, 1] [, 2][1, ] 1 3[2, ] 4 11

Inner product> x = c(1, 7,6, 4)> y = c(2,2, 1, 5)> x[1] 1 7 6 4> y[1] 2 2 1 5> t(x)% %y

[, 1][1, ] 2> sum(x y)[1] 2> crossprod(x, y)

[, 1][1, ] 2

Order or dimension of a vector> y = c(2,2, 1, 5)> y[1] 2 2 1 5> length(y)[1] 4

Length of a vector> ynorm = sqrt(crossprod(y, y))> ynorm

[, 1][1, ] 5.830952

Elementwise multiplication> a = matrix(c(3, 6, 2, 1), 2, 2, byrow = T )> a

[, 1] [, 2][1, ] 3 6[2, ] 2 1> b = matrix(c(7,4,3, 2), 2, 2, byrow = T )> b

[, 1] [, 2][1, ] 7 4[2, ] 3 2> a b

[, 1] [, 2][1, ] 21 24[2, ] 6 2

Kronecker product> a = matrix(c(2, 4, 0,2, 3,1), 3, 2, byrow = T )> a

[, 1] [, 2][1, ] 2 4[2, ] 0 2[3, ] 3 1> b = matrix(c(5, 3, 2, 1), 2, 2, byrow = T )> b

[, 1] [, 2][1, ] 5 3[2, ] 2 1> kronecker(a, b)

[, 1] [, 2] [, 3] [, 4][1, ] 10 6 20 12[2, ] 4 2 8 4[3, ] 0 0 10 6[4, ] 0 0 4 2[5, ] 15 9 5 3[6, ] 6 3 2 1

What happens when the dimensions of the matrices orvectors are not appropriate for the operation

> a = matrix(c(1, 1, 1, 2), 2, 2, byrow = T )> b = matrix(c(3, 0,1, 1,1, 4), 2, 3, byrow = T )> a

[, 1] [, 2][1, ] 1 1[2, ] 1 2> b

[, 1] [, 2] [, 3][1, ] 3 0 1[2, ] 1 1 4> a+ b

Error in a + b : non-conformable arrays

Create a diagonal matrix

> I3 = diag(rep(1, 3))> I3

[, 1] [, 2] [, 3][1, ] 1 0 0[2, ] 0 1 0[3, ] 0 0 1

Trace of a matrix

> a = matrix(c(1, 3, 7, 2, 4, 8, 5, 6, 9), 3, 3, byrow = T )> a

[, 1] [, 2] [, 3][1, ] 1 3 7[2, ] 2 4 8[3, ] 5 6 9> tra = sum(diag(a))> tra

[1]14

Inverse of a matrix

> ainv = solve(a)> ainv[r] [, 1] [, 2] [, 3][1, ] 6 7.5 2[2, ] 11 13.0 3[3, ] 4 4.5 1> a% %ainv[r] [, 1] [, 2] [, 3][1, ] 1.000000e+ 00 0 1.776357e 15[2, ] 7.105427e 15 1 1.776357e 15[3, ] 0.000000e+ 00 0 1.000000e+ 00

Rank of a matrix

> A = matrix(c(1, 1, 1, 2, 5,1, 0, 1,1), 3, 3, byrow = T )> A

[, 1] [, 2] [, 3][1, ] 1 1 1[2, ] 2 5 1[3, ] 0 1 1> qr(A)$rank

[1] 2

rank(AA) = rank(A)

> AtA = t(A)% %A> AtA

[, 1] [, 2] [, 3][1, ] 5 11 1[2, ] 11 27 5[3, ] 1 5 3> qr(AtA)$rank[1] 2> solve(AtA)Error in solve.default(AtA) :

system is computationally singular: reciprocal condition number =

2.58191e-18

Row/column operations:

> sum(A)[1] 9> apply(A, 1, sum)[1] 3 6 0> apply(A, 2, sum)[1] 3 7 1> apply(A, 1, prod)[1] 1 10 0> apply(A, 1,mean)[1] 1 2 0> apply(A, 1, var)[1] 0 9 1

Eigenvalues and Eigenvectors

> A = matrix(c(1, 2,1, 4), 2, 2, byrow = T )

> A[, 1] [, 2]

[1, ] 1 2[2, ] 1 4

> EA = eigen(A)> EA$values[1] 3 2

$vectors[, 1] [, 2]

[1, ] 0.7071068 0.8944272[2, ] 0.7071068 0.4472136

Singular Value Decomposition

> A = matrix(c(2, 0, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1), 3, 4, byrow = T )> A

[, 1] [, 2] [, 3] [, 4][1, ] 2 0 1 1[2, ] 0 2 1 1[3, ] 1 1 1 1> svdA = svd(A)> svdA$d[1]3.464102 2.000000 0.000000

$u[, 1] [, 2] [, 3]

[1, ] 0.5773503 7.071068e 01 0.4082483[2, ] 0.5773503 7.071068e 01 0.4082483[3, ] 0.5773503 1.110223e 16 0.8164966

$v[, 1] [, 2] [, 3]

[1, ] 0.5 7.071068e 01 0.3535534[2, ] 0.5 7.071068e 01 0.3535534[3, ] 0.5 1.110223e 16 0.8535534[4, ] 0.5 1.110223e 16 0.1464466

> svdA$u% %t(svdA$u)[, 1] [, 2] [, 3]

[1, ] 1.000000e+ 00 8.326673e 17 1.110223e 16[2, ] 8.326673e 17 1.000000e+ 00 1.110223e 16[3, ] 1.110223e 16 1.110223e 16 1.000000e+ 00

> t(svdA$v)% %svdA$v[, 1] [, 2] [, 3]

[1, ] 1.000000e+ 00 1.110223e 16 1.387779e 17[2, ] 1.110223e 16 1.000000e+ 00 6.027326e 17[3, ] 1.387779e 17 6.027326e 17 1.000000e+ 00

> svdA$u% %diag(svdA$d)% %t(svdA$v)[, 1] [, 2] [, 3] [, 4]

[1, ] 2.000000e+ 00 1.110223e 16 1 1[2, ] 2.220446e 16 2.000000e+ 00 1 1[3, ] 1.000000e+ 00 1.000000e+ 00 1 1

> diag(svdA$d)% %diag(svdA$d)[, 1] [, 2] [, 3]

[1, ] 12 0 0[2, ] 0 4 0[3, ] 0 0 0

> eigen(A% %t(A))$values[1] 1.20000e+ 01 4.00000e+ 00 1.24345e 14> eigen(t(A)% %A)$values[1] 1.200000e+ 01 4.000000e+ 00 8.881784e 15 0.000000e+ 00

Relationship between trace/determinanat andeigenvalues

> B = matrix(c(1, 1,2,1, 2, 1, 0, 1,1), 3, 3, byrow = T )> B

[, 1] [, 2] [, 3][1, ] 1 1 2[2, ] 1 2 1[3, ] 0 1 1

> eigenB = eigen(B)> eigenB$values[1] 2 1 1$vectors

[, 1] [, 2] [, 3][1, ] 0.3015113 0.8017837 7.071068e 01[2, ] 0.9045340 0.5345225 1.559729e 17[3, ] 0.3015113 0.2672612 7.071068e 01

> traceB = sum(diag(B))> traceB[1] 2> Re(sum(eigenB$values))[1] 2

> detB = det(B)> detB[1] 2> Re(prod(eigenB$values))[1] 2

Eigenvalues of a square symmetric matrix

> A = matrix(c(4, 2,1, 2, 4,4,1,4, 8), 3, 3, byrow = T )> A

[, 1] [, 2] [, 3][1, ] 4 2 1[2, ] 2 4 4[3, ] 1 4 8

> EA = eigen(A)> EA$values[1] 11 4 1$vectors

[, 1] [, 2] [, 3][1, ] 0.2672612 0.8728716 0.4082483[2, ] 0.5345225 0.2182179 0.8164966[3, ] 0.8017837 0.4364358 0.4082483

> SV DA = svd(A)> SV DA$d[1] 11 4 1$u

[, 1] [, 2] [, 3][1, ] 0.2672612 0.8728716 0.4082483[2, ] 0.5345225 0.2182179 0.8164966[3, ] 0.8017837 0.4364358 0.4082483$v

[, 1] [, 2] [, 3][1, ] 0.2672612 0.8728716 0.4082483[2, ] 0.5345225 0.2182179 0.8164966[3, ] 0.8017837 0.4364358 0.4082483

An example of a square symmetric matrix that is notpositive definite

> W = matrix(c(4, 2,1, 2, 6,4,1,4,9), 3, 3, byrow = T )> W

[, 1] [, 2] [, 3][1, ] 4 2 1[2, ] 2 6 4[3, ] 1 4 9

> EW = eigen(W )> EW$values[1] 8.151345 2.865783 10.017128$vectors

[, 1] [, 2] [, 3][1, ] 0.4665008 0.88381658 0.0352886[2, ] 0.8550024 0.46079428 0.2379907[3, ] 0.2266009 0.08085101 0.9706262

> SV DW = svd(W )> SV DW$d[1] 10.017128 8.151345 2.865783$u

[, 1] [, 2] [, 3][1, ] 0.0352886 0.4665008 0.88381658[2, ] 0.2379907 0.8550024 0.46079428[3, ] 0.9706262 0.2266009 0.08085101$v

[, 1] [, 2] [, 3][1, ] 0.0352886 0.4665008 0.88381658[2, ] 0.2379907 0.8550024 0.46079428[3, ] 0.9706262 0.2266009 0.08085101

Inverse of a matrix obtained by using the spectraldecomposition

> A = matrix(c(4, 2, 2, 6), 2, 2, byrow = T )> A

[, 1] [, 2][1, ] 4 2[2, ] 2 6> Ainv = solve(A)> Ainv

[, 1] [, 2][1, ] 0.3 0.1[2, ] 0.1 0.2> Aev = eigen(A)$vectors> Aeval = eigen(A)$values

> Ainv2 = Aev% %diag(1/Aeval)% %t(Aev)> Ainv2

[, 1] [, 2][1, ] 0.3 0.1[2, ] 0.1 0.2

Solutions to linear equations

> x = c(3, 5)> x[1] 3 5

> b = solve(A, x)> b

[1] 0.4 0.7