Matrix

Matrix

Wanho Choi (wanochoi.com)

• A branch of mathematics

concerning linear equations using vector and matrix

Linear Algebra (

선형대수학, 線型代數學)

System of Linear Eqns.

(

연립 일차 방정식, 聯立一次方程式)

• 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}

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 + ⋯ + x_n2 A ∙ B = a₁ × b₁ + a₂ × b₂ + ⋯ + a_n × b_n n × 1 1 × n

Matrix-Matrix Addition

• Component-wise addition a₁₁ + b₁₁ a₁₂ + b₁₂ ⋯ a_1n + b_1n a₂₁ + b₂₁ a₂₂ + b₂₂ ⋯ a_2n + b_2n ⋮ ⋮ ⋱ ⋮ a_m1 + b_m1 a_m2+ b_m2 ⋯ a_mn + b_mn = a₁₁ a₁₂ ⋯ a_1n a₂₁ a₂₂ ⋯ a_2n ⋮ ⋮ ⋱ ⋮ a_m1 a_m2 ⋯ a_mn + b₁₁ b₁₂ ⋯ b_1n b₂₁ b₂₂ ⋯ b_2n ⋮ ⋮ ⋱ ⋮ b_m1 b_m2 ⋯ b_mn

C

_m×n

= A

_m×n

× B

_m×n

• 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

Matrix-Matrix Multiplication

c

_ij

=

_∑

n k=0

a

_ik

× b

_kj c₁₁ c₁₂ ⋯ c_1n c₂₁ c₂₂ ⋯ c_2n ⋮ ⋮ ⋱ ⋮ c_m1 c_m2 ⋯ c_mn = a₁₁ a₁₂ ⋯ a_1p a₂₁ a₂₂ ⋯ a_2p ⋮ ⋮ ⋱ ⋮ a_m1 a_m2 ⋯ a_mp b₁₁ b₁₂ ⋯ b_1n b₂₁ b₂₂ ⋯ b_2n ⋮ ⋮ ⋱ ⋮ b_p1 b_p2 ⋯ b_pn

C

_m×n

= A

_m×p

× B

_p×n

Outer Product Matrix

ABT = a₁ a₂ ⋮ a_n [b1 b2 ⋯ bn] = a₁b₁ a₁b₂ ⋯ a₁b_n a₂b₁ a₂b₂ ⋯ a₂b_n ⋮ ⋮ ⋱ ⋮ a_nb₁ a_nb₂ ⋯ a_nb_n A ∙ B = ATB = a₁ × b₁ + a₂ × b₂ + ⋯ + a_n × b_n n × 1 1 × n n × n 1 × n n × 1 1 × 1

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 a_ij = 0 if i ≠ j

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

a_ij = 0 a_ij = 0 if i ≠ j a_ij = 1 if i = j a_ij = a_ji B = AT m = n m = n a_ij = 0 if i > j a_ij = 0 if i < j a_ij = − a_ji b_ij = a_ji