Circulant Matrix Recall:Permutation A permutation p on {1,2,...,n}is even:if p can be restored to natural order by an even number of interchanges. 命电有这女子 Matrix Theory Matrices -7/29
Circulant Matrix Recall: Permutation A permutation p on {1, 2, . . . , n} is ▸ even: if p can be restored to natural order by an even number of interchanges. ▸ odd: if p can be restored to natural order by an odd number of interchanges. Matrix Theory Matrices - 7/29
Circulant Matrix Recall:Permutation A permutation p on (1,2,...,n is even:if p can be restored to natural order by an even number of interchanges. odd:if p can be restored to natural order by an odd number of interchanges. 奇电有这头 Matrix Theory Matrices -7/29
Circulant Matrix Recall: Permutation A permutation p on {1, 2, . . . , n} is ▸ even: if p can be restored to natural order by an even number of interchanges. ▸ odd: if p can be restored to natural order by an odd number of interchanges. Matrix Theory Matrices - 7/29
Circulant Matrix Example of permutation For instance,consider the permutations on {1,2,3,4). ·p=(2,1,4,3)is even -after interchanging 2 and 1,4 and 3(two interchanges) =→(1,2,3,4) 命电有这女 Matrix Theory Matrices -8/29
Circulant Matrix Example of permutation For instance, consider the permutations on {1, 2, 3, 4}. ▸ p = (2, 1, 4, 3) is even –after interchanging 2 and 1, 4 and 3 (two interchanges) Ô⇒ (1, 2, 3, 4) ▸ p = (1, 4, 3, 2) is odd –after interchanging 4 and 2 Ô⇒ (1, 2, 3, 4) Matrix Theory Matrices - 8/29
Circulant Matrix Example of permutation For instance,consider the permutations on {1,2,3,4). p=(2,1,4,3)is even -after interchanging 2 and 1,4 and 3(two interchanges) =→(1,2,3,4) p=(1,4,3,2)is odd -after interchanging 4 and 2=(1,2,3,4) 命电有这女子 Matrix Theory Matrices -8/29
Circulant Matrix Example of permutation For instance, consider the permutations on {1, 2, 3, 4}. ▸ p = (2, 1, 4, 3) is even –after interchanging 2 and 1, 4 and 3 (two interchanges) Ô⇒ (1, 2, 3, 4) ▸ p = (1, 4, 3, 2) is odd –after interchanging 4 and 2 Ô⇒ (1, 2, 3, 4) Matrix Theory Matrices - 8/29
Circulant Matrix Permutation matrix An orthogonal matrix that can reorder the rows or columns of a matrix is called a permutation matrix. 命电有这女子 Matrix Theory Matrices -9/29
Circulant Matrix Permutation matrix ▸ An orthogonal matrix that can reorder the rows or columns of a matrix is called a permutation matrix. ▸ In mathematics, in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry 1 in each row and each column and 0s elsewhere. ▸ Each such matrix represents a specific permutation of m elements and, when used to multiply another matrix, can produce that permutation in the rows or columns of the other matrix. ▸ It is an identity matrix whose rows have been reordered (permuted). ▸ One can also think of a permutation matrix as an identity matrix whose columns have been reordered. Matrix Theory Matrices - 9/29