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1 and 2 are called a ring because they have the + and · operations defined on them since they are square matrices.
Due to associativity of multiplication in #Theorem 1.3.2, we do not write parentheses around multiplication:
Plays the role of "1" in multiplication:
(In other words, 1s along diagonal, 0s everywhere else) ...such that
Acts as an identity vector:
, where is a vector of 0s, except the th element is 1.
Plays role of 0 in addition:
If for matrices and , we say than is the inverse of : .
Not every matrix is invertible. A matrix that does not have a multiplicative inverse is said to be singular.
Transpose of a Matrix
For a matrix , the transpose of , written , will be and is defined as follows:
Geometrically, the first row becomes the first column, second row becomes second column, etc. The matrix is simply "reflected" about its "diagonal"
- (opposite order)
A matrix is said to be symmetric iff . This means that
A diagonal matrix is a square () matrix that has values along its diagonal ()
A special form of diagonal matrices is when (i.e. all numbers along diagonal are the same): these special diagonal matrices commute with arbitrary matrices
Matrices and Graphs
A graph consists of vertices (data points) and edges that connect them.
A graph with entries ( through ) can be represented by a adjacency matrix:
First, let be the (, ) entry of .
represents the number of walks of length from to .
For all and for all , the indicated operations are defined:
- (commutativity of addition)
- (associativity of addition)
- (associativity of multiplication)
- (right distributivity)
- (left distributivity)
If and are nonsingular matrices, then is also nonsingular and
(note the opposite order on the right-hand side)
For nonsingular matrices , is also nonsingular and