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 m rows and n columns.
Common types of matrices that we encounter in finite elements are:
diagonal elements (aii) nonzero.
with each of its nonzero elements (aii) equal to 1.
such that aij = aji.
such that aij = − aji.
Note that the diagonal elements of a skew-symmetric matrix have to be zero: .
Let and be two matrices with components aij and bij, respectively. Then
Let be a matrix with components aij and let λ be a scalar quantity. Then,
Let be a matrix with components aij. Let be a matrix with components bij.
The product is defined only if n = p. The matrix is a matrix with components cij. Thus,
Similarly, the product is defined only if q = m. The matrix is a matrix with components dij. 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 aij. Then the transpose of the matrix is defined as the matrix with components bij = aji. That is,
A important identity involving the transpose of matrices is
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 minors as
In short, the determinant of a matrix has the value
where Mij is the determinant of the submatrix of formed by eliminating row i and column j from .
Some useful identities involving the determinant are given below.
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.
determinant of its inverse.
is equal to the original matrix.
We usually use numerical methods such as Gaussian elimination to compute the inverse of a matrix.
A thorough explanation of this material can be found at Eigenvalue, eigenvector and eigenspace. However, for further study, let us consider the following examples:
Which vector is an eigenvector for ?
We have , and
Thus, is an eigenvector.
We have that since , is not an eigenvector for