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# Matrix

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## Matrix

Wanho Choi (wanochoi.com)

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• A branch of mathematics

concerning linear equations using vector and matrix

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### 연립 일차 방정식, 聯立一次方程式)

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• A rectangular array of numbers with

dimensions m (# of rows) by n (# of columns).

ij

m×n

11

12

1n

21

22

2n

m1

m2

### ⋯ a

mn i ∈ {1,2,⋯, m} j ∈ {1,2,⋯, n}

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### N-Dimensional Vector

• It can be thought as an n×1 column matrix:

T

1

2

n

1

2

### x

n X = x12 + x22 + ⋯ + xn2 A ∙ B = a1 × b1 + a2 × b2 + ⋯ + an × bn n × 1 1 × n

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• Component-wise addition a11 + b11 a12 + b12 ⋯ a1n + b1n a21 + b21 a22 + b22 ⋯ a2n + b2n ⋮ ⋮ ⋱ ⋮ am1 + bm1 am2+ bm2 ⋯ amn + bmn = a11 a12 ⋯ a1n a21 a22 ⋯ a2n ⋮ ⋮ ⋱ ⋮ am1 am2 ⋯ amn + b11 b12 ⋯ b1n b21 b22 ⋯ b2n ⋮ ⋮ ⋱ ⋮ bm1 bm2 ⋯ bmn

m×n

m×n

### × B

m×n

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• The new vector is the dot product of each

row of the matrix with the column vector.

i

n k=0

ik

k

11

12

1n

21

22

2n

m1

m2

mn

1

2

n

1

2

n

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ij

n k=0

ik

### × b

kj c11 c12 ⋯ c1n c21 c22 ⋯ c2n ⋮ ⋮ ⋱ ⋮ cm1 cm2 ⋯ cmn = a11 a12 ⋯ a1p a21 a22 ⋯ a2p ⋮ ⋮ ⋱ ⋮ am1 am2 ⋯ amp b11 b12 ⋯ b1n b21 b22 ⋯ b2n ⋮ ⋮ ⋱ ⋮ bp1 bp2 ⋯ bpn

m×n

m×p

p×n

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### Outer Product Matrix

ABT = a1 a2an [b1 b2 ⋯ bn] = a1b1 a1b2 ⋯ a1bn a2b1 a2b2 ⋯ a2bn ⋮ ⋮ ⋱ ⋮ anb1 anb2 ⋯ anbn A ∙ B = ATB = a1 × b1 + a2 × b2 + ⋯ + an × bn n × 1 1 × n n × n 1 × n n × 1 1 × 1

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### Types of Matrix

Square Matrix

Diagonal Matrix Sparse Matrix

Null Matrix (Zero Matrix) Identity Matrix

Transpose Matrix Symmetric Matrix

Skew Symmetric Matrix Upper Triangular Matrix Lower Triangular Matrix

m = n m = n m = n m = n m = n aij = 0 if i ≠ j

# of zero elements ≫ # of non-zero-elements

aij = 0 aij = 0 if i ≠ j aij = 1 if i = j aij = aji B = AT m = n m = n aij = 0 if i > j aij = 0 if i < j aij = − aji bij = aji

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