M a t r i c es
矩阵DEFINITION 1,
A matrix is a rectangular array of numbers,A matrix
with m rows and n columns is called an m × n matrix,
The plural of matrix is matrices,A matrix with the
same number of rows as columns is called equal if they
have the same number of rows and the same number of
columns and the corresponding entries in every position
are equal.
M a t r i c es
矩阵DEFINITION 2,
Let
The (i,j)th element or entry of A is the element aij,that is,
the number in the ith row and jth column of A,A
convenient shorthand notation for expressing the matrix
A is to write A = [aij],which indicates that A is the matrix
with its (i,j)th element equal to aij.
The ith row of A is the 1 × n
matrix [ai1,ai2,…,a in],The
jth column of A is the n × 1
matrix
M a t r i c es
矩阵DEFINITION 3,
Let A = [aij] and B = [bij] be m × n matrices,The
sum of A and B,denoted by A + B,is the m × n matrix
that has aij + bij as its (i,j)th element,In other words,A
+ B = [aij +bij].
M a t r i c es
矩阵DEFINITION 4,
Let A be an m × k matrix and B be a k × n
matrix,The product of A and B,denoted by AB,is
the m × n matrix with (i,j)th entry equal to the sum
of the products of the corresponding elements from
the ith row of A and the jth column of B,In AB =
[cij],then
Cij = ai1b1j + ai2b2j + … + a ikbkj =
M a t r i c es
矩阵DEFINITION 5,
The identity matrix of order n is the n × n matrix In
= [ ],where =1 if i = j and = 0 if i ≠ j,Hence
M a t r i c es
矩阵DEFINITION 6,
Let A = [aij] be an m × n matrix,The transqose of
A,denoted by At,is the n × m matrix obtained by
interchanging the rows and columns of A,In other
words,if At = [bij],then bij = aji for I = 1,2,…,n and
j = 1,2,…,m.
M a t r i c es
矩阵DEFINITION 7,
A square matrix A is called symmetric if A = At,
Thus A = [aij] is symmetric if aij = aji for all i and j with
1≤i≤n and 1≤j≤n.
M a t r i c es
矩阵DEFINITION 8,
Let A = [aij] and B = [bij] be m × n zero-one matrices,
Then the join of A and B is the zero-one matrix with (i,
j)th entry aij∨ bij,The join of A and B is denoted by
A∨ B,The meet of A and B is the zero-one matrix with
(i,j)th entry aij∧ bij,The meet of A and B is denoted by
A∧ B.
M a t r i c es
矩阵DEFINITION 9,
Let A = [aij] be an m × k zero-one matrix and B = [bij]
be a k × n zero-one matrix,Then the Boolean product
of A and B,denoted by A⊙ B,is the m × n matrix with
(i,j)th entry [cij] where
cij = (ai1∧ b1j)∨ (ai2∧ b2j)∨ … ∨ (aik∧ bkj).
M a t r i c es
矩阵
Let A be a square zero-one matrix and let r be a
positive integer,The rth Boolean power of A is the
Boolean product of r A,The rth Boolean product of A is
denoted by A[r],Hence
A[r] = A⊙ A⊙ A⊙ A⊙ A⊙ A⊙ A⊙ A
(This is well defined since the Boolean product of
matrices is associative.) We also define A[0] to be In.
DEFINITION 10,
R times
矩阵DEFINITION 1,
A matrix is a rectangular array of numbers,A matrix
with m rows and n columns is called an m × n matrix,
The plural of matrix is matrices,A matrix with the
same number of rows as columns is called equal if they
have the same number of rows and the same number of
columns and the corresponding entries in every position
are equal.
M a t r i c es
矩阵DEFINITION 2,
Let
The (i,j)th element or entry of A is the element aij,that is,
the number in the ith row and jth column of A,A
convenient shorthand notation for expressing the matrix
A is to write A = [aij],which indicates that A is the matrix
with its (i,j)th element equal to aij.
The ith row of A is the 1 × n
matrix [ai1,ai2,…,a in],The
jth column of A is the n × 1
matrix
M a t r i c es
矩阵DEFINITION 3,
Let A = [aij] and B = [bij] be m × n matrices,The
sum of A and B,denoted by A + B,is the m × n matrix
that has aij + bij as its (i,j)th element,In other words,A
+ B = [aij +bij].
M a t r i c es
矩阵DEFINITION 4,
Let A be an m × k matrix and B be a k × n
matrix,The product of A and B,denoted by AB,is
the m × n matrix with (i,j)th entry equal to the sum
of the products of the corresponding elements from
the ith row of A and the jth column of B,In AB =
[cij],then
Cij = ai1b1j + ai2b2j + … + a ikbkj =
M a t r i c es
矩阵DEFINITION 5,
The identity matrix of order n is the n × n matrix In
= [ ],where =1 if i = j and = 0 if i ≠ j,Hence
M a t r i c es
矩阵DEFINITION 6,
Let A = [aij] be an m × n matrix,The transqose of
A,denoted by At,is the n × m matrix obtained by
interchanging the rows and columns of A,In other
words,if At = [bij],then bij = aji for I = 1,2,…,n and
j = 1,2,…,m.
M a t r i c es
矩阵DEFINITION 7,
A square matrix A is called symmetric if A = At,
Thus A = [aij] is symmetric if aij = aji for all i and j with
1≤i≤n and 1≤j≤n.
M a t r i c es
矩阵DEFINITION 8,
Let A = [aij] and B = [bij] be m × n zero-one matrices,
Then the join of A and B is the zero-one matrix with (i,
j)th entry aij∨ bij,The join of A and B is denoted by
A∨ B,The meet of A and B is the zero-one matrix with
(i,j)th entry aij∧ bij,The meet of A and B is denoted by
A∧ B.
M a t r i c es
矩阵DEFINITION 9,
Let A = [aij] be an m × k zero-one matrix and B = [bij]
be a k × n zero-one matrix,Then the Boolean product
of A and B,denoted by A⊙ B,is the m × n matrix with
(i,j)th entry [cij] where
cij = (ai1∧ b1j)∨ (ai2∧ b2j)∨ … ∨ (aik∧ bkj).
M a t r i c es
矩阵
Let A be a square zero-one matrix and let r be a
positive integer,The rth Boolean power of A is the
Boolean product of r A,The rth Boolean product of A is
denoted by A[r],Hence
A[r] = A⊙ A⊙ A⊙ A⊙ A⊙ A⊙ A⊙ A
(This is well defined since the Boolean product of
matrices is associative.) We also define A[0] to be In.
DEFINITION 10,
R times
