MATH347

MATH347 L4: More matrix operations

New concepts:
- Matrix transpose
- Block matrices
- Block matrix addition, multiplication, transposition
- Combined operations:
  - Addition and multiplication (Matrix distributivity)
  - Transposition and addition
  - Transposition and multiplication

Matrix transpose

Definition. Given a matrix $𝑨 \in ℝ^{m \times n}$ ,

$𝑨 = [\begin{array}{llll} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{array}]$

the transpose of $𝑨$ , denoted as $𝑨^{T}$ $\in ℝ^{n \times m}$ is

$𝑨^{T} = [\begin{array}{llll} a_{11} & a_{21} & \dots & a_{n 1} \\ a_{12} & a_{22} & \dots & a_{n 2} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{1 m} & a_{2 m} & \dots & a_{n m} \end{array}] .$

${[𝑨^{T}]}_{i, j} = {[𝑨]}_{j, i}$

Transposition switches row and column vectors

Matrix transpose examples

>>	A=[1 2 3; 4 5 6; 7 8 9]; disp(A)

1 2 3 4 5 6 7 8 9

>>	transpose(A)

$(\begin{array}{ccc} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{array})$

>>

A'

$(\begin{array}{ccc} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{array})$

>>	u=[1 2 3]; u'

$(\begin{array}{c} 1 \\ 2 \\ 3 \end{array})$

Block matrices

$𝑨 \in ℝ^{4 \times 4}$ is a mtrix with 4 rows and 4 columns
$𝑨 = [\begin{array}{llll} 1 & 2 & - 1 & 0 \\ 2 & 1 & 0 & 2 \\ - 1 & 0 & 1 & 2 \\ 0 & 2 & 2 & 1 \end{array}], 𝑫 = [\begin{array}{llll} 1 & - 1 & 1 & 0 \\ - 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & - 1 \\ 0 & 1 & - 1 & 1 \end{array}]$
It is useful to recognize common structures in a matrix
$𝑨 = [\begin{array}{ll} 𝑩 & 𝑪 \\ 𝑪 & 𝑩 \end{array}], 𝑩 = [\begin{array}{ll} 1 & 2 \\ 2 & 1 \end{array}], 𝑪 = [\begin{array}{ll} - 1 & 0 \\ 0 & 2 \end{array}]$ $𝑫 = [\begin{array}{ll} 𝑬 & 𝑭 \\ 𝑭 & 𝑬 \end{array}], 𝑬 = [\begin{array}{ll} 1 & - 1 \\ - 1 & 1 \end{array}], 𝑭 = [\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}]$

Block matrix addition, multiplication, transposition

Matrix block addition
$𝑨 = [\begin{array}{ll} 𝑩 & 𝑪 \\ 𝑪 & 𝑩 \end{array}], 𝑫 = [\begin{array}{ll} 𝑬 & 𝑭 \\ 𝑭 & 𝑬 \end{array}],$ $𝑨 + 𝑫 = [\begin{array}{ll} 𝑩 & 𝑪 \\ 𝑪 & 𝑩 \end{array}] + [\begin{array}{ll} 𝑬 & 𝑭 \\ 𝑭 & 𝑬 \end{array}] = [\begin{array}{ll} 𝑩 + 𝑬 & 𝑪 + 𝑭 \\ 𝑪 + 𝑭 & 𝑩 + 𝑬 \end{array}]$
Matrix block multiplication
$𝑨 𝑫 = [\begin{array}{ll} 𝑩 & 𝑪 \\ 𝑪 & 𝑩 \end{array}] [\begin{array}{ll} 𝑬 & 𝑭 \\ 𝑭 & 𝑬 \end{array}] = [\begin{array}{ll} 𝑩 𝑬 + 𝑪 𝑭 & 𝑩 𝑭 + 𝑪 𝑬 \\ 𝑪 𝑬 + 𝑩 𝑭 & 𝑪 𝑭 + 𝑩 𝑬 \end{array}]$
Matrix block transposition
$𝑴 = [\begin{array}{ll} 𝑼 & 𝑽 \\ 𝑿 & 𝒀 \end{array}], 𝑴^{T} = [\begin{array}{ll} 𝑼^{T} & 𝑿^{T} \\ 𝑽^{T} & 𝒀^{T} \end{array}]$

Transpose of matrix columns

$𝑨 \in ℝ^{m \times n}$ contains $n$ column vectors with $m$ components each
$𝑨 = [\begin{array}{llll} 𝒂_{1} & 𝒂_{2} & \dots & 𝒂_{n} \end{array}]$
The transpose switches rows and columns
$𝑨^{T} = [\begin{array}{l} 𝒂_{1}^{T} \\ 𝒂_{2}^{T} \\ ⋮ \\ 𝒂_{n}^{T} \end{array}] \in ℝ^{n \times m}$
has $n$ rows and $m$ columns

Block operation examples: forming blocks, block assembly

>>	A=[1 2 -1 0; 2 1 0 2; -1 0 1 2; 0 2 2 1]; disp(A)

1 2 -1 0 2 1 0 2 -1 0 1 2 0 2 2 1

>>	B=A(1:2,1:2); C=A(1:2,3:4); [B C; C B]

$(\begin{array}{cccc} 1 & 2 & - 1 & 0.0 \\ 2 & 1 & 0.0 & 2 \\ - 1 & 0.0 & 1 & 2 \\ 0.0 & 2 & 2 & 1 \end{array})$

>>	D=[1 -1 1 0; -1 1 0 1; 1 0 1 -1; 0 1 -1 1];

>>	E=D(1:2,1:2); F=D(1:2,3:4); [D [E F; F E]]

$(\begin{array}{cccccccc} 1 & - 1 & 1 & 0.0 & 1 & - 1 & 1 & 0.0 \\ - 1 & 1 & 0.0 & 1 & - 1 & 1 & 0.0 & 1 \\ 1 & 0.0 & 1 & - 1 & 1 & 0.0 & 1 & - 1 \\ 0.0 & 1 & - 1 & 1 & 0.0 & 1 & - 1 & 1 \end{array})$

Block operation examples: addition

$𝑨 + 𝑫 = [\begin{array}{ll} 𝑩 & 𝑪 \\ 𝑪 & 𝑩 \end{array}] + [\begin{array}{ll} 𝑬 & 𝑭 \\ 𝑭 & 𝑬 \end{array}] = [\begin{array}{ll} 𝑩 + 𝑬 & 𝑪 + 𝑭 \\ 𝑪 + 𝑭 & 𝑩 + 𝑬 \end{array}]$

>>

A+D

$(\begin{array}{cccc} 2 & 1 & 0.0 & 0.0 \\ 1 & 2 & 0.0 & 3 \\ 0.0 & 0.0 & 2 & 1 \\ 0.0 & 3 & 1 & 2 \end{array})$

>>	[B C; C B] + [E F; F E]

$(\begin{array}{cccc} 2 & 1 & 0.0 & 0.0 \\ 1 & 2 & 0.0 & 3 \\ 0.0 & 0.0 & 2 & 1 \\ 0.0 & 3 & 1 & 2 \end{array})$

>>	[B+E C+F; C+F B+E]

$(\begin{array}{cccc} 2 & 1 & 0.0 & 0.0 \\ 1 & 2 & 0.0 & 3 \\ 0.0 & 0.0 & 2 & 1 \\ 0.0 & 3 & 1 & 2 \end{array})$

Block operation examples: multiplication

$𝑨 𝑫 = [\begin{array}{ll} 𝑩 & 𝑪 \\ 𝑪 & 𝑩 \end{array}] [\begin{array}{ll} 𝑬 & 𝑭 \\ 𝑭 & 𝑬 \end{array}] = [\begin{array}{ll} 𝑩 𝑬 + 𝑪 𝑭 & 𝑩 𝑭 + 𝑪 𝑬 \\ 𝑪 𝑬 + 𝑩 𝑭 & 𝑪 𝑭 + 𝑩 𝑬 \end{array}]$

>>

A*D

$(\begin{array}{cccc} - 2 & 1 & 0.0 & 3 \\ 1 & 1 & 0.0 & 3 \\ 0.0 & 3 & - 2 & 1 \\ 0.0 & 3 & 1 & 1 \end{array})$

>>	[BE+CF BF+CE; CE+BF CF+BE]

$(\begin{array}{cccc} - 2 & 1 & 0.0 & 3 \\ 1 & 1 & 0.0 & 3 \\ 0.0 & 3 & - 2 & 1 \\ 0.0 & 3 & 1 & 1 \end{array})$

>>

Block operation examples: transposition

$𝑴 = [\begin{array}{ll} 𝑼 & 𝑽 \\ 𝑿 & 𝒀 \end{array}], 𝑴^{T} = [\begin{array}{ll} 𝑼^{T} & 𝑿^{T} \\ 𝑽^{T} & 𝒀^{T} \end{array}]$

>>	M=[1 2 3 4; 5 6 7 8; -1 -2 -3 -4; -5 -6 -7 -8]; disp([M M'])

1 2 3 4 1 5 -1 -5 5 6 7 8 2 6 -2 -6 -1 -2 -3 -4 3 7 -3 -7 -5 -6 -7 -8 4 8 -4 -8

>>	U=M(1:2,1:2); V=M(1:2,3:4); X=M(3:4,1:2); Y=M(3:4,3:4);

>>	[U' X'; V' Y']

$(\begin{array}{cccc} 1 & 5 & - 1 & - 5 \\ 2 & 6 & - 2 & - 6 \\ 3 & 7 & - 3 & - 7 \\ 4 & 8 & - 4 & - 8 \end{array})$

>>

Combined operations

Distributivity $𝑨 \in ℝ^{m \times n}, 𝑩, 𝑪 \in ℝ^{n \times p}, 𝑨 (𝑩 + 𝑪) = 𝑨 𝑩 + 𝑨 𝑪$
Transposition of sum ${(𝑩 + 𝑪)}^{T} = 𝑩^{T} + 𝑪^{T}$
Transposition of scaling $α \in ℝ$ , ${(α 𝑨)}^{T} = α 𝑨^{T}$
Transposition of product: Start by matrix-vector product for $𝒙 \in ℝ^{n \times 1}$
$𝑨 𝒙 = (\begin{array}{ccc} 𝒂_{1} & \dots & 𝒂_{n} \end{array}) 𝒙 = x_{1} 𝒂_{1} + \dots + x_{n} 𝒂_{n} \Rightarrow$ ${(𝑨 𝒙)}^{T} = x_{1} 𝒂_{1}^{T} + \dots + x_{n} 𝒂_{n}^{T} = [\begin{array}{lll} x_{1} & \dots & x_{n} \end{array}] [\begin{array}{l} 𝒂_{1}^{T} \\ ⋮ \\ 𝒂_{n}^{T} \end{array}] = 𝒙^{T} 𝑨^{T}$
Transposition of product ${(𝑨 𝑩)}^{T} = 𝑩^{T} 𝑨^{T}$
${(𝑨 𝑩)}^{T} = {(\begin{array}{ccc} 𝑨 𝒃_{1} & \dots & 𝑨 𝒃_{n} \end{array})}^{T} = (\begin{array}{c} {(𝑨 𝒃_{1})}^{T} \\ ⋮ \\ {(𝑨 𝒃_{n})}^{T} \end{array}) = (\begin{array}{c} 𝒃_{1}^{T} 𝑨^{T} \\ ⋮ \\ 𝒃_{n}^{T} 𝑨^{T} \end{array}) = (\begin{array}{c} 𝒃_{1}^{T} \\ ⋮ \\ 𝒃_{n}^{T} \end{array}) 𝑨^{T}$