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Matrices and Vectors
3 Matrices and Vectors
This Section is a very concise introduction to the algebra of matrices and
vectors. This will let us
study mathematical models of the types we saw in the preceding illustrative
examples in a more
general, unified fashion.
3.1 Matrices
A (real) matrix of size m *n is an array of mn real numbers arranged in m rows
and n columns:
The n*m matrix A^{T} obtained by exchanging rows
and columns of A is called the transpose of
A. A matrix A is said to be symmetric if A = A^{T} .
The sum of two matrices of equal size is the matrix of the entrybyentry sums,
and the scalar
product of a real number a and an m *n matrix A is the m* n matrix of all the
entries of A, each
multiplied by a . The difference of two matrices of equal size A and B is
A  B = A + (1)B :
The product of an m *p matrix A and a p* n matrix B is an m* n matrix C with
entries
Examples
Let
Then,
and
The product AB is not defined, because A and B have incompatible sizes. Furthermore,
and
3.2 Vectors
A (real) ndimensional vector is an ntuple of real numbers
v = (v_{1},..., v_{n}) .
There is a natural, onetoone correspondence between ndimensional vectors and
n *1 matrices:
The matrix on the right is called the column vector
corresponding to the vector on the left.
There is also a natural, onetoone correspondence between ndimensional vectors
and 1* n
matrices:
The matrix on the right is called the row vector corresponding to the vector on
the left.
If a is a vector, then the symbol a also denotes the corresponding column
vector, so that the
corresponding row vector is a^{T} .
All algebraic operations on vectors are inherited from the corresponding matrix
operations,
when defined. In addition , the inner product of two ndimensional vectors
a = (a_{1},..., a_{n}) and b = (b_{1},..., b_{n})
is the real number equal to the matrix product a^{T}b. It is easy to
verify that this is also equal to
b^{T} a. Two vectors that have a zero inner product are said to be
orthogonal.
The norm of a vector a is
obviously a nonnegative number . A unit vector is a vector
with norm one.
Examples
The vector a = (2, 1, 0) corresponds to row vector a^{T} = [2, 1, 0]
and to column vector
3.2 Vectors
The inner product of a and b = (1, 0, 1) is
and the norm of a is
The vector
has unit norm:
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