A matrix is a two dimensional rectangular array of quantities or expressions in rows and columns that is treated as a single entity and manipulated according to particular rules. Often these elements will be numbers.
Matrices can be considered as an (ordered) collection of vectors. Specifically, an m by n matrix is equivalent to n Euclidean column vectors of dimension m.
Similarly, we may also consider row vectors, so that an m by n matrix is equivalent to m Euclidean row vectors of dimension n.
denotes a matrix, where aij is the entry at the ith row and jth column (the ij-entry).
can be considered as three 2-dimensional column vectorsA = [a1 a2 a3], a1 = [ 1 ], a2 = [ 2 ], a3 = [ 3 ] 4 5 6
ExampleIt can also be considered as two 3-dimensional row vectors. A = [ b1 ], b1 = (1, 2, 3), b2 = (4, 5, 6) b2
For example, for three column matrices A = [a1 a2 a3], B = [b1 b2 b3], we haveA + B = [a1 + b1 a2 + b2 a3 + b3], cA = [ca1 ca2 ca3].
Use a comma to separate columns.
For an n by n square matrix A = (aij), the terms a11, a22, ..., nn are the diagonal entries of the matrix. A is a diagonal matrix if all the off-diagonal entries are zero. In other words, aij = 0 for i >< j. The following are all the 2 by 2 and 3 by 3 diagonal matrices.