Circulant Matrix Circulant matrix generated from a vector as the first row 2 Cn-2 Cn-3 .0 色电这女子 Matrix Theory Matrices -6/29
Circulant Matrix Circulant matrix generated from a vector as the first row ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ c0 c1 c2 . . . cn−1 cn−1 c0 c1 . . . cn−2 cn−2 cn−1 c0 . . . cn−3 ⋮ ⋮ ⋮ ⋮ ⋮ c1 c2 c3 . . . c0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ Example 1: Example 2: primary permutation matrix Matrix Theory Matrices - 6/29 C = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 2 3 4 5 5 1 2 3 4 4 5 1 2 3 3 4 5 1 2 2 3 4 5 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ▸ x = [1, 2, 3, 4, 5]; C = gallery( ′ circul ′ , x); C = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 1 0 . . . 0 0 0 1 . . . 0 ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 0 . . . 1 1 0 0 . . . 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Circulant Matrix Circulant matrix generated from a vector as the first row a Cn-1 Cn-2 Cn-3 . C3 Co Example 1: C= 154 2154 3 3215 43215 5432 2 3 4 1 色电这女子 Matrix Theory Matrices -6/29
Circulant Matrix Circulant matrix generated from a vector as the first row ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ c0 c1 c2 . . . cn−1 cn−1 c0 c1 . . . cn−2 cn−2 cn−1 c0 . . . cn−3 ⋮ ⋮ ⋮ ⋮ ⋮ c1 c2 c3 . . . c0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ Example 1: Example 2: primary permutation matrix Matrix Theory Matrices - 6/29 C = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 2 3 4 5 5 1 2 3 4 4 5 1 2 3 3 4 5 1 2 2 3 4 5 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ▸ x = [1, 2, 3, 4, 5]; C = gallery( ′ circul ′ , x); C = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 1 0 . . . 0 0 0 1 . . . 0 ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 0 . . . 1 1 0 0 . . . 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Circulant Matrix Circulant matrix generated from a vector as the first row C2 Cn-1 G Cn-2 N Cn-3 C2 C3 Co 5 ×=[1,2,3,4,5] Example 1: 15 C=gallery('circul',x); C= 4 2154 321 3 5 4321 432 2 34 5 1 色电有这女了 Matrix Theory Matrices -6/29
Circulant Matrix Circulant matrix generated from a vector as the first row ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ c0 c1 c2 . . . cn−1 cn−1 c0 c1 . . . cn−2 cn−2 cn−1 c0 . . . cn−3 ⋮ ⋮ ⋮ ⋮ ⋮ c1 c2 c3 . . . c0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ Example 1: Example 2: primary permutation matrix Matrix Theory Matrices - 6/29 C = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 2 3 4 5 5 1 2 3 4 4 5 1 2 3 3 4 5 1 2 2 3 4 5 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ▸ x = [1, 2, 3, 4, 5]; C = gallery( ′ circul′ , x); C = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 1 0 . . . 0 0 0 1 . . . 0 ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 0 . . . 1 1 0 0 . . . 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Circulant Matrix Circulant matrix generated from a vector as the first row 0 a Cn-1 Cn-2 Cn-3 C3 Co Example 2: 0 1 0 0 0 1 0 C= 0 0 0 10 0 0 色电这女了 Matrix Theory Matrices -6/29
Circulant Matrix Circulant matrix generated from a vector as the first row ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ c0 c1 c2 . . . cn−1 cn−1 c0 c1 . . . cn−2 cn−2 cn−1 c0 . . . cn−3 ⋮ ⋮ ⋮ ⋮ ⋮ c1 c2 c3 . . . c0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ Example 1: Example 2: primary permutation matrix Matrix Theory Matrices - 6/29 C = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 2 3 4 5 5 1 2 3 4 4 5 1 2 3 3 4 5 1 2 2 3 4 5 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ▸ x = [1, 2, 3, 4, 5]; C = gallery( ′ circul ′ , x); C = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 1 0 . . . 0 0 0 1 . . . 0 ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 0 . . . 1 1 0 0 . . . 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Circulant Matrix Circulant matrix generated from a vector as the first row Co C1 C2 。。 Cn-1 co G Cn-2 Cn-2 Cn-1 co Cn-3 C2 C3 Co Example 2:primary permutation matrix 010 0 0 01 0 C= 0 0 0 1 10 0 0 色电这女子 Matrix Theory Matrices -6/29
Circulant Matrix Circulant matrix generated from a vector as the first row ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ c0 c1 c2 . . . cn−1 cn−1 c0 c1 . . . cn−2 cn−2 cn−1 c0 . . . cn−3 ⋮ ⋮ ⋮ ⋮ ⋮ c1 c2 c3 . . . c0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ Example 1: Example 2: primary permutation matrix Matrix Theory Matrices - 6/29 C = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 2 3 4 5 5 1 2 3 4 4 5 1 2 3 3 4 5 1 2 2 3 4 5 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ▸ x = [1, 2, 3, 4, 5]; C = gallery( ′ circul ′ , x); C = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 1 0 . . . 0 0 0 1 . . . 0 ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 0 . . . 1 1 0 0 . . . 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠