In mathematics, a matrix (plural matrices) is a rectangular array
The individual items in an m × n matrix A, often denoted by ai,j, where max i = m and max j = n, are called its elements or entries. Provided that they have the same size (each matrix has the same number of rows and the same number of columns as the other).
Two matrices can be added or subtracted element by element (see Conformable matrix). The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second (i.e., the inner dimensions are the same, n for Am,n × Bn,p). Any matrix can be multiplied element-wise by a scalar from its associated field.
A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three-dimensional space is a linear transformation, which can be represented by a rotation matrix R: if v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two transformation matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a linear transformation is obtainable (along with other information) from the matrix's eigenvalues and eigenvectors.
Common types of matrices that we encounter in finite elements are:
- a row vector that has one row and columns.
- a column vector that has rows and one column.
- a square matrix that has an equal number of rows and columns.
- a diagonal matrix which is a square matrix with only the
diagonal elements () non zero.
- the identity matrix () which is a diagonal matrix and
with each of its non zero elements () equal to 1.
- a symmetric matrix which is a square matrix with elements
such that .
- a skew-symmetric matrix which is a square matrix with elements
such that .
Note that the diagonal elements of a skew-symmetric matrix have to be zero: .
The determinant of a matrix is defined only for square matrices.
For a matrix , we have
For a matrix, the determinant is calculated by expanding into
In short, the determinant of a matrix has the value
where is the determinant of the sub-matrix of formed by eliminating row and column from .
Some useful identities involving the determinant are given below.
- If is a matrix, then
- If is a constant and is a matrix, then
- If and are two matrices, then
If you think you understand determinants, take the quiz.
Let and be two matrices with components
and , respectively. Then
Multiplication by a scalar[edit | edit source]
Let be a matrix with components and let
be a scalar quantity. Then,
Multiplication of matrices[edit | edit source]
Let be a matrix with components . Let be a matrix with components .
The product is defined only if . The matrix is a matrix with components . Thus,
Similarly, the product is defined only if . The matrix is a matrix with components . We have
Clearly, in general, i.e., the matrix product is not commutative.
However, matrix multiplication is distributive. That means
The product is also associative. That means
Let be a matrix with components . Then the transpose of the matrix is defined as the matrix with components . That is,
An important identity involving the transpose of matrices is
Let be a matrix. The inverse of is denoted by and is defined such that
where is the identity matrix.
The inverse exists only if . A singular matrix
does not have an inverse.
An important identity involving the inverse is
since this leads to:
Some other identities involving the inverse of a matrix are given below.
- The determinant of a matrix is equal to the multiplicative inverse of the
determinant of its inverse.
- The determinant of a similarity transformation of a matrix
is equal to the original matrix.
We usually use numerical methods such as Gaussian elimination to compute
the inverse of a matrix.
Eigenvalues and eigenvectors[edit | edit source]
A thorough explanation of this material can be found at Eigenvalue, eigenvector and eigenspace. However, for further study, let us consider the following examples:
- Let :
Which vector is an eigenvector for ?
Thus, is an eigenvector.
- Is an eigenvector for ?
We have that since , is not an eigenvector for