Matrix Matlab

Mean, Variance, Covariance

μ = E[X]

σ^2 = var[X] = E[(X - E[X])^2]

X =[x_11 x_12 x_13 ; x_21 x_22 x_23 ; x_31 x_32 x_33]

X = ({{X_11, X_12, X_13}, {X_21, X_22, X_23}, {X_31, X_32, X_33}})

m = mean (x) = 1/nUnderoverscript[∑, i = 1, arg3][x_i1 x_i2 … x_in]

mean (x) = 1/3Underoverscript[∑, i = 1, arg3][x_i1 x_i2 x_i3] =[(x_11 + x_21 + x_31)/3 (x_12 + x_22 + x_32)/3 (x_13 + x_23 + x_33)/3]

var(X,1) = mean(v.^2)

v = X - ones (3, 1) * mean (x)

var (X, 1) = mean (v .^2)

cov (X) = (v ' * v)/(n - 1)

[n, p] = size (X)

v = X - ones (3, 1) * mean (x)

cov (X) = (v ' * v)/(n - 1)

var (x) = diag (cov (x)) '

Matrix Mathematica

X = {-1, 4, -9, -0, 3, 3, 9, -2, 5, 5} ;

Mean[X] = Total[X]/Length[X]

Variance[X] = Total[(X - Mean[X])^2]/(Length[X] - 1)

StandardDeviation[X] = Sqrt[Variance[X]]

Mean

X = ({{-1, 4, -9}, {0, -3, 3}, {9, -2, 5}}) ;

Mean[X]

( {{8/3}, {-1/3}, {-1/3}} )

Covariance

V = (X - {Mean[X], Mean[X], Mean[X]})

V . V/(Length[X] - 1)

( {{91/3, -29/3, 85/3}, {-29/3, 43/3, -83/3}, {85/3, -83/3, 172/3}} )

Covariance to Variance : var(x)

V = (X - {Mean[X], Mean[X], Mean[X]})

Total[V . V/(Length[X] - 1) * IdentityMatrix[3]]

Variance[X]

{91/3, 43/3, 172/3}

Mean[(X - {Mean[X], Mean[X], Mean[X]})^2] * Length[X]/(Length[X] - 1)

Variance : var(x,1)

Mean[(X - {Mean[X], Mean[X], Mean[X]})^2]

{20.22,9.55,38.22}

IdentityMatrix[3]

DiagonalMatrix[{a,b,c,d}]

X[[1]]

{-1,4,-9}

X[[All, 1]]

{-1,0,9}

Inverse, Determinant, Trace, Transpose

Inverse[({{a, b}, {c, d}})] = ( {{d/(-b c + a d), -b/(-b c + a d)}, {-c/(-b c + a d), a/(-b c + a d)}} )

Det[({{a, b}, {c, d}})] = -b c + a d

Tr[({{a, b}, {c, d}})] = a + d

({{a, b}, {c, d}})  = ( {{a, c}, {b, d}} )

Mean[({{a, b}, {c, d}})] = ( {{1/2 (a + c)}, {1/2 (b + d)}} )

Variance[({{a, b}, {c, d}})] = ( {{1/2 (a - c)^2}, {1/2 (b - d)^2}} )

Covariance[({{a, b}, {c, d}})] = ( {{1/2 (a - c)^2, 1/2 (a - c) (b - d)}, {1/2 (a - c) (b - d), 1/2 (b - d)^2}} )

How to Covariance

X = ({{a, b}, {c, d}})

Covariance[X_] := {V = (X - {Mean[X], Mean[X]}) ; R = V . V/(Length[X] - 1)//Simplify//MatrixForm}[[1]] ;

eigenvalue & eigenvector

eigenvalue : λ

eigenvector : V_λ

A V_λ = λ V_λ

A = V_λ λ V_λ^(-1)

(A - λ Eye) V_λ = 0

Det[A - λ Eye] = 0

2×2 Matrix

A = ({{a, b}, {c, d}})

Eye = ({{1, 0}, {0, 1}})

Det[( {{a - λ, b}, {c, d - λ}} )] = (a - λ) (d - λ) - b c = λ^2 - (a + d) λ + (a d - b c)

Solve[Det[A - λ Eye] == 0, λ][[1]][[1]][[2]]

λ = 1/2 (a + d± (4 b c + (a - d)^2)^(1/2))

Tr[A] = a + d

Det[A] = a d - b c

Det[A - λ Eye] = λ^2 - Tr[A] λ + Det[A]

λ = 1/2 ((a + d) ± ((a + d)^2 - 4 (a d - b c))^(1/2))

= 1/2 (Tr[A] ± (Tr[A]^2 - 4 Det[A])^(1/2))

Matlab

[V, D] = eig (A)

A = V * D * inv (V)

Similarity Transformation Diagonalizable

A V_1 = λ_1V_1, …, A V_n = λ_nV_n

C =[V_1, …, V_n]

(1×n) Vector

A C = A[V_1, …, V_n] =[A V_1, …, A V_n] =[λ_1V_1, …, λ_nV_n]

C^(-1) A C = ({{λ_1, ···, 0}, {:, ·., :}, {0, ···, λ_n}})

if C is orthogonal (직교)

then C^(-1) = C

S = ({{1, 1}, {1, 0}})

S^(-1) = Inverse[S] = ( {{0, 1}, {1, -1}} )

A = ({{-1, 4}, {0, 3}})

S^(-1) A S = Inverse[S] . A . S = ({{3, 0}, {0, -1}})