(MATH) A square or rectangular
array of numbers, usually written enclosed in a large pair of parentheses.
Matrices, which are added and multiplied using a special set of rules,
are extremely useful for representing quantities, particularly in some
branches of physics. A matrix can be thought of as a linear operator
on vectors. Matrix-vector multiplication
can be used to describe geometric transformations such as scaling, rotation, reflection, and translation.
|A general 3×3 matrix
The first mathematical use of the word "matrix" was around 1850 by James Sylvester who saw a matrix as a way
of obtaining determinants, but didn't
fully appreciate their potential. Within a year of his first use of
the term, he introduced the idea to Arthur Cayley who was the first to publish the inverse of a matrix and to treat matrices as purely abstract mathematical forms.
The use of mathematical arrays to solve problems predates the application
of the name by about 2000 years. Around 200 BC in the Chinese text Juizhang Suanshu (Nine Chapters on the Mathematical
Arts) the author solves a system of three equations in three unknowns
by placing the coefficients on a
counting board and solving by a process that today would be called Gaussian
elimination. "Matrix" comes from the same Latin root that gives
us "mother", and was used to refer to the womb and to pregnant
animals. It became generalized to mean any situation or substance that
contributes to the origin of something.
A is described as an m × n matrix
over F, where F is a field,
without characteristic, to which all of the mn elements of A belong. A is made up of m1
× n matrices (row vectors) and nm × 1 matrices
The main diagonal In the square matrix [aij],
the elements a11, a22, ..., ann,
running from the top left corner to the bottom right corner.
Types of matrix
A diagonal matrix is
a matrix that has 0 entries along all nondiagonal entries, i.e., only the main diagonal may have non-zero values.
A unimodular matrix is a square matrix whose determinant is 1.
A Hankel matrix is one in which all the elements are the same along any diagonal that slopes from northeast to southwest.
A Jordan matrix (or Jordan block), is one whose diagonal elements are all equal (and nonzero) and whose elements above the principal diagonal are equal to 1, but all other elements are 0.
A Toeplitz matrix is one in which all the elements are the
same along any diagonal that slopes from northwest to southeast.
If two matrices A and B of F are both m × n, the result of their addition
is defined to be the m × n matrix of which a
typical element (aij + bij).
Their addition is thus commutative and associative. Moreover, the set
of m × n matrices of F forms an Abelian
group under addition since it has an identity element (see group), O, the zero matrix whose elements are all zero, since A + O = A for all A, and since every A has an inverse = -A, whose typical element is (-aij),
such that A + (-A) = O.
If l is an element of F, the product lA is defined as the matrix whose typical element is laij.
Multiplication of one matrix by another is possible only if the first
has the same number of columns as the second has rows. If A is an m × n and B an n × p matrix then their product AB is an m × p matrix with typical element
ai1b1j + ai2b2j + ... + ainbnj
Notice that, in this case, BA does not exist unless p = m since otherwise B does not have the
same number of columns as A has rows.
Transposition and symmetry
The transpose of an m × n matrix A is an n × m matrix A', obtained
by setting the row vectors of A as the column vectors
of A', the column vectors of A as
the row vectors of A'. Some basic results emerge:
(A')' = A,
where k, l, are elements of F. A matrix A is termed symmetric when A = A' and
skew-symmetric when A = -A': of course,
only matrices where n = m can be symmetric or skew-symmetric.
For symmetry the element aij in A must equal aji for all i, j; for
skew-symmetry aij + aji = 0,
for all i, j. See also determinant.
(AB)' = B'A',
and (kA + lB)' = kA' + lB',
- (GEOLOGY) The solid matter in
which a fossil or crystal is embedded. Also, a binding substance (e.g., cement in concrete).
AND PLANETARY SCIENCE