Nonlinear finite elements/Matrices

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.

MatricesEdit

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 ExercisesEdit

Practice: Expressing Linear Equations As Matrices

Types of MatricesEdit

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 additionEdit

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

 

Multiplication by a scalarEdit

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

 

Multiplication of matricesEdit

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 matrixEdit

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 matrixEdit

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 matrixEdit

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 eigenvectorsEdit

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