Numerical Methods Contents - SAM

1.1.2 Matrices
Matrices = two-dimensional arrays of real/complex numbers
⎛
A := ⎝ a ⎞
11 ... a 1m
. . ⎠ ∈ K n,m , n, m ∈ N .
a n1 ... a nm
vector space of n × m-matrices: (n ˆ= number of rows, m ˆ= number of columns)
⎛
Zero matrix: O = ⎝ 0 . .. 0
⎞
. ... . ⎠ ∈ K n,m , MATLAB: O = zeros(n,m);
0 . .. 0
⎛
Diagonal matrix: D = ⎝ d ⎞
1 0
. .. ⎠ ∈ K n,n , MATLAB: D = diag(d); with vector d
0 d n
✎ notation: bold capital roman letters, e.g., A,S,Y
MATLAB:
K n,1 ↔ column vectors, K 1,n ↔ row vectors
✄ vectors are 1 × n/n × 1-matrices
✄ initialization:
Accessing matrix entries & sub-matrices (✎
⎛
A := ⎝ a ⎞
11 . .. a 1m
. . ⎠
a n1 . .. a nm
⎛
A = [1,2;3,4;5,6]; → 3 × 2 matrix ⎝ 1 2
⎞
3 4⎠ .
5 6
notations):
→ entry (A) i,j = a ij , 1 ≤ i ≤ n, 1 ≤ j ≤ m ,
→ i-th row, 1 ≤ i ≤ n: a i,: = (A) i,:
Ôº¾ ½º½
,
→ j-th column, 1 ≤ j ≤ m: a i,: = (A) :,j ,
1 ≤ k ≤ l ≤ n ,
→ matrix block (a ij ) i=k,...,l = (A) k:l,r:s ,
1 ≤ r ≤ s ≤ m .
MATLAB: Matrix A ↦→ entry at position (i,j) = A(i,j)
↦→ i-th row = A(i,:)
↦→ j-th column = A(:,j)
↦→
matrix block
(sub-matrix)
(a ij ) i=k,...,l
j=r,...,s
= (A) k:l,r:s = A(k:l,r:s)
Transposed matrix:
⎛
A T = ⎝ a ⎞T
⎛
11 ... a 1m
. . ⎠ := ⎝ a ⎞
11 . .. a n1
. . ⎠ ∈ K m,n .
a n1 ... a nm a 1m . .. a mn
Adjoint matrix (Hermitian transposed):
⎛
A H := ⎝ a ⎞H
⎛ ⎞
11 . .. a 1m ... ā n1
. . ⎠ := ⎝ā11 . . ⎠ ∈ K m,n .
a n1 . .. a nm ā 1m ... ā mn
✎ notation: ā ij = Re(a ij ) − iIm(a ij ) complex conjugate of a ij .
Special matrices:
⎛
Identity matrix: I = ⎝ 1 0
⎞
Ôº¾ ½º½
... ⎠ ∈ K n,n , MATLAB: I = eye(n);
0 1
Remark 1.1.1 (Matrix storage formats). (for dense/full matrices, cf. Sect. 2.6)
A ∈ K m,n
⎛
A = ⎝ 1 2 3
⎞
4 5 6⎠
7 8 9
linear array (size mn) + index computations
(Note:
leading dimension (row major, column major))
Row major (C-arrays, bitmaps, Python):
A_arr 1 2 3 4 5 6 7 8 9
Access to entry a ij of A ∈ K n,m , i = 1,...,n,
j = 1, ...,m:
row major:
column major:
a ij ↔A_arr(m*(i-1)+(j-1))
a ij ↔A_arr(n*(j-1)+(i-1))
Column major (Fortran, MATLAB, OpenGL):
A_arr 1 4 7 2 5 8 3 6 9
Example 1.1.2 (Impact of data access patterns on runtime).
row major
Cache hierarchies ❀ slow access of “remote” memory sites !
column oriented
A = randn(n,n);
for j = 1:n-1,
A(:,j+1) = A(:,j+1) - A(:,j);
end
access
row oriented
A = randn(n);
for i = 1:n-1,
A(i+1,:) = A(i+1,:) - A(i,:);
end
column major
Ôº¾ ½º½
2 K = 3 ; res = [ ] ;
access
n = 3000 ❀ 0.1s n = 3000 ❀ 0.3s
Code 1.1.3: timing for row and column oriented matrix access in MATLAB
1 % Timing for row/column operations
△
Ôº¾ ½º½