Continuum mechanics/Matrices

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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.

Matrices[edit]

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 Exercises[edit]

Practice: Expressing Linear Equations As Matrices

Types of Matrices[edit]

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: .

Matrix addition[edit]

Let and be two matrices with components and , respectively. Then

Multiplication by a scalar[edit]

Let be a matrix with components and let be a scalar quantity. Then,

Multiplication of matrices[edit]

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[edit]

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[edit]

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 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[edit]

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]

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  ?

We have , and

Thus, is an eigenvector.

  • Is an eigenvector for  ?

We have that since , is not an eigenvector for