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@WolfMoon

知っていて当たり前-10　行列

Posted at

知っていて当たり前-10　行列

行列のもっとも簡単な定義法。

A = [1 2 3;
     4 5 6]

2×3 Matrix{Int64}:
 1  2  3
 4  5  6

1.　行列のサイズ

row, col = size(A) # (行数, 列数) のタプルを返す

(2, 3)

2.　各次元のサイズ

size(A, 1) # 行数

2

size(A, 2) # 列数

3

3.　全要素数

length(A)

6

4.　次元数

ndims(A)

2

5.　零行列

zeros(2, 3) # 型はデフォルトで Float64

2×3 Matrix{Float64}:
 0.0  0.0  0.0
 0.0  0.0  0.0

zeros(Int, 2, 3) # 第 1 引数で型を指定できる

2×3 Matrix{Int64}:
 0  0  0
 0  0  0

6.　1 行列

ones(2, 3)

2×3 Matrix{Float64}:
 1.0  1.0  1.0
 1.0  1.0  1.0

ones(Bool, 2, 3)

2×3 Matrix{Bool}:
 1  1  1
 1  1  1

7.　Bool 行列

trues(2,3)

2×3 BitMatrix:
 1  1  1
 1  1  1

falses(2, 3)

2×3 BitMatrix:
 0  0  0
 0  0  0

8.　すべて§同じ任意の要素を持つ行列 fill() 

f0 = fill(5, 2, 3)

2×3 Matrix{Int64}:
 5  5  5
 5  5  5

9.　既存の配列と型・次元数が同じ，初期化されていない配列 similar() 

f2 = similar(f0) # Python の like

2×3 Matrix{Int64}:
          0  4985017360  4985434720
 4985434720           0  4999865536

f3 = similar(f0, Float64) # 型を指定することもできる

2×3 Matrix{Float64}:
 0.0           2.46293e-314  2.46313e-314
 2.46313e-314  0.0           2.46293e-314

10.　型・次元を指定した初期化されていない配列を定義する

Array{Float64}(undef, 2, 3)

2×3 Matrix{Float64}:
 2.22963e-314  2.22963e-314  2.22963e-314
 2.22963e-314  2.22963e-314  2.22963e-314

11.　乱数配列

rand(2, 3) # 一様乱数

2×3 Matrix{Float64}:
 0.462003  0.399307  0.421369
 0.152579  0.111431  0.657385

randn(2, 3) # 標準正規乱数

2×3 Matrix{Float64}:
 -0.877612   0.859351  -1.2645
  1.20194   -0.281498   0.181104

12.　単位行列

using LinearAlgebra
Array{Int}(I, 3, 3)

3×3 Matrix{Int64}:
 1  0  0
 0  1  0
 0  0  1

必ずしも正方行列である必要はない（Array と Matix は同義語）。

Matrix{Float64}(I, 3, 5)

3×5 Matrix{Float64}:
 1.0  0.0  0.0  0.0  0.0
 0.0  1.0  0.0  0.0  0.0
 0.0  0.0  1.0  0.0  0.0

Matrix(Diagonal(ones(Int, 3)))

3×3 Matrix{Int64}:
 1  0  0
 0  1  0
 0  0  1

13.　対角行列

13.1.　任意の要素を持つ対角行列の作成

using LinearAlgebra
diagm(0 => 1:3)

3×3 Matrix{Int64}:
 1  0  0
 0  2  0
 0  0  3

13.2.　対角要素の上に任意の要素を持つ行列の作成

diagm(1 => 1:3)

4×4 Matrix{Int64}:
 0  1  0  0
 0  0  2  0
 0  0  0  3
 0  0  0  0

13.3.　対角要素の下に任意の要素を持つ行列の作成

diagm(-1 => 1:3)

4×4 Matrix{Int64}:
 0  0  0  0
 1  0  0  0
 0  2  0  0
 0  0  3  0

結果を切り詰めることもできる。

diagm(-1 => 1:3)[2:end, :]

3×4 Matrix{Int64}:
 1  0  0  0
 0  2  0  0
 0  0  3  0

複数指定することができる

diagm(-1 => 1:3, 1 => [10, 20, 30, 40])

5×5 Matrix{Int64}:
 0  10   0   0   0
 1   0  20   0   0
 0   2   0  30   0
 0   0   3   0  40
 0   0   0   0   0

Diagonal([1,2,3])

3×3 Diagonal{Int64, Vector{Int64}}:
 1  ⋅  ⋅
 ⋅  2  ⋅
 ⋅  ⋅  3

Diagonal(ones(3))

3×3 Diagonal{Float64, Vector{Float64}}:
 1.0   ⋅    ⋅ 
  ⋅   1.0   ⋅ 
  ⋅    ⋅   1.0

Diagonal(ones(Int, 3))

3×3 Diagonal{Int64, Vector{Int64}}:
 1  ⋅  ⋅
 ⋅  1  ⋅
 ⋅  ⋅  1

### 13.4.　対角要素の取り出し

a = [1 2 3; 4 5 6; 7 8 9]

3×3 Matrix{Int64}:
 1  2  3
 4  5  6
 7  8  9

using LinearAlgebra
diag(a)

3-element Vector{Int64}:
 1
 5
 9

13.5.　対角の一つ上の要素の取り出し

diag(a, 1)

2-element Vector{Int64}:
 2
 6

13.6.　対角の一つ下の要素の取り出し

diag(a, -1)

2-element Vector{Int64}:
 4
 8

13.7.　行列の対角成分に代入

function diag!(a, b)
    n, m = size(a)
    [a[i, j] = b[i, j] for j=1:n, i=1:n if i == j]
end

a = [1 2 3; 4 5 6; 7 8 9]
b = [10 20 30; 40 50 60; 70 80 90]
diag!(a, b);

a

3×3 Matrix{Int64}:
 10   2   3
  4  50   6
  7   8  90

14.　上三角要素，下三角要素の抽出

A = [3 2 1;
     2 4 1;
     1 1 5]

3×3 Matrix{Int64}:
 3  2  1
 2  4  1
 1  1  5

14.1.　対角成分を含む下三角要素をベクトルにして取り出す

using LinearAlgebra
A[tril(trues(size(A)))]

6-element Vector{Int64}:
 3
 2
 1
 4
 1
 5

14.2.　対角成分の k 下の三角要素をベクトルにして取り出す

tril(trues(size(A)), -1)

3×3 BitMatrix:
 0  0  0
 1  0  0
 1  1  0

triu(trues(size(A)), 2)

3×3 BitMatrix:
 0  0  1
 0  0  0
 0  0  0

A[tril(trues(size(A)), -1)]

3-element Vector{Int64}:
 2
 1
 1

14.3.　対角成分を含む上三角要素をベクトルにして取り出す

A[triu(trues(size(A)))]

6-element Vector{Int64}:
 3
 2
 4
 1
 1
 5

14.4.　下三角行列

a = [1 2 3; 4 5 6; 7 8 9]

3×3 Matrix{Int64}:
 1  2  3
 4  5  6
 7  8  9

14.5.　下三角行列を取り出す

using LinearAlgebra
LowerTriangular(a)　# using LinearAlgebra

3×3 LowerTriangular{Int64, Matrix{Int64}}:
 1  ⋅  ⋅
 4  5  ⋅
 7  8  9

14.6.　下三角行列へ代入

function lowertri!(a, b)
    n, m = size(a)
    [a[i, j] = b[i, j] for j=1:n, i=1:n if i > j]
end

lowertri! (generic function with 1 method)

a = [1 2 3; 4 5 6; 7 8 9]
b = [11 12 13; 14 15 16; 17 18 19]
lowertri!(a, b)

3-element Vector{Int64}:
 14
 17
 18

a

3×3 Matrix{Int64}:
  1   2  3
 14   5  6
 17  18  9

15.　転置

a = [1 2 3; 4 5 6; 7 8 9]

3×3 Matrix{Int64}:
 1  2  3
 4  5  6
 7  8  9

a'

3×3 adjoint(::Matrix{Int64}) with eltype Int64:
 1  4  7
 2  5  8
 3  6  9

transpose(a)

3×3 transpose(::Matrix{Int64}) with eltype Int64:
 1  4  7
 2  5  8
 3  6  9

16.　回転 rotr90(), rotl90() 

a = [1 2 3; 4 5 6]

2×3 Matrix{Int64}:
 1  2  3
 4  5  6

rotr90(a)

3×2 Matrix{Int64}:
 4  1
 5  2
 6  3

rotr90(rotr90(a))

2×3 Matrix{Int64}:
 6  5  4
 3  2  1

rotr90(rotr90(rotr90(a)))

3×2 Matrix{Int64}:
 3  6
 2  5
 1  4

rotr90(a, 3)

3×2 Matrix{Int64}:
 3  6
 2  5
 1  4

rotl90(a)

3×2 Matrix{Int64}:
 3  6
 2  5
 1  4

17.　回転シフト

a = [1 2 3; 4 5 6; 7 8 9];
circshift(a, (0, 1))

3×3 Matrix{Int64}:
 3  1  2
 6  4  5
 9  7  8

circshift(a, 1)

3×3 Matrix{Int64}:
 7  8  9
 1  2  3
 4  5  6

18.　ランク rank() 

a = [1 2 3; 4 5 6; 7 8 9]
using LinearAlgebra
rank(a)

2

19.　行列式 det() 

using LinearAlgebra
det(a)

0.0

トレース tr()

using LinearAlgebra
tr(a)

15

クロネッカー積 cron()

a = [1 2 3
    4 5 6]

2×3 Matrix{Int64}:
 1  2  3
 4  5  6

b = [1 2
     3 6
     7 11]

3×2 Matrix{Int64}:
 1   2
 3   6
 7  11

kron(a, b)

6×6 Matrix{Int64}:
  1   2   2   4   3   6
  3   6   6  12   9  18
  7  11  14  22  21  33
  4   8   5  10   6  12
 12  24  15  30  18  36
 28  44  35  55  42  66

20.　逆行列 inv()

逆行列を明示的に使うのは避けるべし。

a = [1.0 3.0 5.0; 6.0 8.0 3.0; 3.0 4.0 2.0]

3×3 Matrix{Float64}:
 1.0  3.0  5.0
 6.0  8.0  3.0
 3.0  4.0  2.0

inv(a)

3×3 Matrix{Float64}:
 -0.8  -2.8   6.2
  0.6   2.6  -5.4
  0.0  -1.0   2.0

inv(a) * a

3×3 Matrix{Float64}:
 1.0  0.0  0.0
 0.0  1.0  0.0
 0.0  0.0  1.0

21.　連立一次方程式 A x = b

a = [1.0 3.0 5.0; 6.0 8.0 3.0; 3.0 4.0 2.0]

3×3 Matrix{Float64}:
 1.0  3.0  5.0
 6.0  8.0  3.0
 3.0  4.0  2.0

b = [3.0, 4.0, 6.0]

3-element Vector{Float64}:
 3.0
 4.0
 6.0

x = inv(a) * b

3-element Vector{Float64}:
  23.599999999999994
 -20.2
   8.0

a * x # ==> b # 誤差が大きい

3-element Vector{Float64}:
 3.0
 3.9999999999999716
 5.999999999999986

a \ b # ==> x = inv(a) * b

3-element Vector{Float64}:
  23.599999999999998
 -20.2
   8.0

a * (a \ b) # 誤差が小さい（ない）

3-element Vector{Float64}:
 3.0
 4.0
 6.0

22.　固有値 eigen(), eigvals(), eigvecs() 

using LinearAlgebra
a = [3 1 5; 2 4 7; 4 2 9]

3×3 Matrix{Int64}:
 3  1  5
 2  4  7
 4  2  9

F = eigen(a)

Eigen{Float64, Float64, Matrix{Float64}, Vector{Float64}}
values:
3-element Vector{Float64}:
  0.5159445156361262
  2.3635607609987486
 13.120494723365105
vectors:
3×3 Matrix{Float64}:
 -0.723466   0.205599  -0.398121
 -0.512966  -0.964335  -0.611863
  0.462019   0.166697  -0.683464

F.values

3-element Vector{Float64}:
  0.5159445156361262
  2.3635607609987486
 13.120494723365105

F.vectors

3×3 Matrix{Float64}:
 -0.723466   0.205599  -0.398121
 -0.512966  -0.964335  -0.611863
  0.462019   0.166697  -0.683464

eigvals(a)

3-element Vector{Float64}:
  0.5159445156361262
  2.3635607609987486
 13.120494723365105

eigvecs(a)

3×3 Matrix{Float64}:
 -0.723466   0.205599  -0.398121
 -0.512966  -0.964335  -0.611863
  0.462019   0.166697  -0.683464

22.1.　固有値の大きい順に並べ替え eigen(..., sortby=x-> -x) 

values, vectors = eigen(a, sortby=x-> -x)

Eigen{Float64, Float64, Matrix{Float64}, Vector{Float64}}
values:
3-element Vector{Float64}:
 13.120494723365105
  2.3635607609987486
  0.5159445156361262
vectors:
3×3 Matrix{Float64}:
 -0.398121   0.205599  -0.723466
 -0.611863  -0.964335  -0.512966
 -0.683464   0.166697   0.462019

23.　ファクトライゼーション

23.1.　一般的な行列の場合の LU 分解 factorize() 

a = [3 1; 2 4;]
F = factorize(a)

LU{Float64, Tridiagonal{Float64, Vector{Float64}}}
L factor:
2×2 Matrix{Float64}:
 1.0       0.0
 0.666667  1.0
U factor:
2×2 Matrix{Float64}:
 3.0  1.0
 0.0  3.33333

F.L

2×2 Matrix{Float64}:
 1.0       0.0
 0.666667  1.0

F.U

2×2 Matrix{Float64}:
 3.0  1.0
 0.0  3.33333

23.2.　対称行列の場合

B = [1.5 2 -4; 2 -1 -3; -4 -3 5]
F = factorize(B)

BunchKaufman{Float64, Matrix{Float64}}
D factor:
3×3 Tridiagonal{Float64, Vector{Float64}}:
 -1.64286   0.0   ⋅ 
  0.0      -2.8  0.0
   ⋅        0.0  5.0
U factor:
3×3 UnitUpperTriangular{Float64, Matrix{Float64}}:
 1.0  0.142857  -0.8
  ⋅   1.0       -0.6
  ⋅    ⋅         1.0
permutation:
3-element Vector{Int64}:
 1
 2
 3

23.3.　SVD 特異値分解 svd() 

A = randn(3, 3)

using LinearAlgebra
U, S, V = svd(A)

SVD{Float64, Float64, Matrix{Float64}}
U factor:
3×3 Matrix{Float64}:
 -0.859406  0.251244    0.445306
  0.462956  0.0127275   0.88629
  0.217007  0.96784    -0.127253
singular values:
3-element Vector{Float64}:
 1.9222248484987037
 1.4673307044588406
 0.1384747325066215
Vt factor:
3×3 Matrix{Float64}:
 -0.758379  -0.0276825   0.651225
 -0.629671   0.289311   -0.720981
  0.168448   0.956835    0.236839

23.4.　QR 分解 qr() 

a = [3 1
     2 4]
qr(a)

LinearAlgebra.QRCompactWY{Float64, Matrix{Float64}}
Q factor:
2×2 LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}}:
 -0.83205  -0.5547
 -0.5547    0.83205
R factor:
2×2 Matrix{Float64}:
 -3.60555  -3.05085
  0.0       2.7735

23.5.　コレスキー分解 cholesky() 

r = [1.000000  0.108015  0.227651
     0.108015  1.000000  0.232754
     0.227651  0.232754  1.000000];
cholesky(r)

Cholesky{Float64, Matrix{Float64}}
U factor:
3×3 UpperTriangular{Float64, Matrix{Float64}}:
 1.0  0.108015  0.227651
  ⋅   0.994149  0.209389
  ⋅    ⋅        0.950963

24.　ベクトル・配列の連結 cat(), vcat(), hcat() 

a = [1,2,3]
b = [10,20,30]
[a; b]

6-element Vector{Int64}:
  1
  2
  3
 10
 20
 30

vcat(a, b)

6-element Vector{Int64}:
  1
  2
  3
 10
 20
 30

cat(a, b, dims=1)

6-element Vector{Int64}:
  1
  2
  3
 10
 20
 30

hcat(a, b)

3×2 Matrix{Int64}:
 1  10
 2  20
 3  30

cat(a, b, dims=2)

3×2 Matrix{Int64}:
 1  10
 2  20
 3  30

x = [1 2 3]
y = [4 5 6 7]
cat(x, y, dims=2)

1×7 Matrix{Int64}:
 1  2  3  4  5  6  7

25.　ブロードキャストとベクトル化 broadcast(), ., @. 

A = [1, 2, 3, 4, 5]

5-element Vector{Int64}:
 1
 2
 3
 4
 5

B = [1 2; 3 4; 5 6; 7 8; 9 10]

5×2 Matrix{Int64}:
 1   2
 3   4
 5   6
 7   8
 9  10

broadcast(+, A, B)

5×2 Matrix{Int64}:
  2   3
  5   6
  8   9
 11  12
 14  15

A .+ B

5×2 Matrix{Int64}:
  2   3
  5   6
  8   9
 11  12
 14  15

sqrt.(B)

5×2 Matrix{Float64}:
 1.0      1.41421
 1.73205  2.0
 2.23607  2.44949
 2.64575  2.82843
 3.0      3.16228

@. sqrt(B) + A - 1.5

5×2 Matrix{Float64}:
 0.5      0.914214
 2.23205  2.5
 3.73607  3.94949
 5.14575  5.32843
 6.5      6.66228

sqrt.(B) .+ A .- 1.5

5×2 Matrix{Float64}:
 0.5      0.914214
 2.23205  2.5
 3.73607  3.94949
 5.14575  5.32843
 6.5      6.66228

## 26.　変換 `map()`

A = [1,2,3]

3-element Vector{Int64}:
 1
 2
 3

map(x -> x^2, A)

3-element Vector{Int64}:
 1
 4
 9

map(A) do x
    x^2
end

3-element Vector{Int64}:
 1
 4
 9

27.　配列の要素を集約 reduce() 

A = [1,2,3]
reduce(+, A)

6

vars = rand(2, 3, 4)
reduce(+, vars, dims=3) # 3次元配列 n x n x k

2×3×1 Array{Float64, 3}:
[:, :, 1] =
 1.685    1.01614  1.66267
 1.01937  2.13548  2.28805

28.　フィルタリング filter() 

A = [1, 2, 3, 4, 5]

5-element Vector{Int64}:
 1
 2
 3
 4
 5

filter(x -> x >= 3, A) # 3 以上の要素を抽出(フィルタリング)

3-element Vector{Int64}:
 3
 4
 5

A[A .>= 3]

3-element Vector{Int64}:
 3
 4
 5

filter(isodd, A) # 奇数要素を抽出

3-element Vector{Int64}:
 1
 3
 5

deleteat!(a, findall(isequal(4), a))

3-element Vector{Int64}:
 1
 2
 3

filter!(x -> x != 4, a)

3-element Vector{Int64}:
 1
 2
 3

29.　ベクトルを配列に reshape() 

vec1 = collect(1:6);

reshape(vec1, 2, 3)

2×3 Matrix{Int64}:
 1  3  5
 2  4  6

reshape(vec1, (2, 3))

2×3 Matrix{Int64}:
 1  3  5
 2  4  6

reshape(vec1, 2, :)

2×3 Matrix{Int64}:
 1  3  5
 2  4  6

reshape(vec1, 6, :)

6×1 Matrix{Int64}:
 1
 2
 3
 4
 5
 6

println(vec1)

[1, 2, 3, 4, 5, 6]

30.　行列をベクトルに vec() 

A = rand(3, 3)

3×3 Matrix{Float64}:
 0.960627    0.688843  0.763275
 0.00694129  0.208644  0.441623
 0.173624    0.105914  0.337504

vec(A)

9-element Vector{Float64}:
 0.9606269419001202
 0.006941291910149472
 0.17362388161403652
 0.6888432524798611
 0.20864371904746726
 0.10591422370536252
 0.7632754965796475
 0.4416234649065267
 0.33750371867325635

A[:]

9-element Vector{Float64}:
 0.9606269419001202
 0.006941291910149472
 0.17362388161403652
 0.6888432524798611
 0.20864371904746726
 0.10591422370536252
 0.7632754965796475
 0.4416234649065267
 0.33750371867325635
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