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Matrix

Wanho Choi (wanochoi.com)

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• A branch of mathematics

concerning linear equations using vector and matrix

Linear Algebra (

선형대수학, 線型代數學)

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System of Linear Eqns.

(

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).

Matrix (

행렬, 行列)

a

ij

: (i, j) element (component, entry)

A

m×n

=

a

11

a

12

⋯ a

1n

a

21

a

22

⋯ a

2n

a

m1

a

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:

X

T

= [x

1

, x

2

, ⋯, x

n

]

X =

x

1

x

2

x

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

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Matrix-Matrix Addition

• 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

C

m×n

= A

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. 

Matrix-Vector Multiplication

b

i

=

n k=0

a

ik

× x

k

a

11

a

12

⋯ a

1n

a

21

a

22

⋯ a

2n

a

m1

a

m2

⋯ a

mn

x

1

x

2

x

n

=

b

1

b

2

b

n

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Matrix-Matrix Multiplication

c

ij

=

n k=0

a

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

C

m×n

= A

m×p

× B

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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