Much of finite elements revolves around forming matrices and
solving systems of linear equations using matrices. This learning resource
gives you a brief review of matrices.
Suppose that you have a linear system of equations
Matrices provide a simple way of expressing these equations. Thus,
we can instead write
An even more compact notation is
Here is a matrix while and are
matrices. In general, an matrix is a set of numbers
arranged in rows and columns.
Practice: Expressing Linear Equations As Matrices
Types of Matrices
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 () nonzero.
- the identity matrix () which is a diagonal matrix and
with each of its nonzero 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: .
Let and be two matrices with components
and , respectively. Then
Multiplication by a scalar
Let be a matrix with components and let
be a scalar quantity. Then,
Multiplication of matrices
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
Transpose of a matrix
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
Determinant of a matrix
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 submatrix 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.
Inverse of a matrix
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
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