The rate at which a medicine spreads through the body can be described by a particular diffusion equation.
The equation is quite complicated, involving several derivatives - with respect to several variables. But solving the equation comes down to finding a diagonal matrix.
If the medicine is an anesthetic, solving the equation gives you information about when it might be safe to operate the patient.
There are multiple other examples of hard problems which can be solved with the same technique. Pretty cool, right?
The diagonalization of a matrix refers to the process of finding three special matrices that produce when multiplied together.
What is special about these matrices is that the matrix in the middle of the multiplication, which we denote , is diagonal, meaning its elements are zero everywhere except along its main diagonal.
This fact means we are not free to choose any three matrices whose product is .
It turns out that the two matrices, other than , will be the inverse of each other and so we formulate the diagonalization as follows:
Such matrices , , and do not always exist, must for instance be a square matrix.
If a matrix is diagonalizable, i.e. the fact that:
can be of great use to us. One example is in instances where we need to perform repeated multiplication with .
Consider the calculation:
and recall what we have previously learned about multiplications with inverse and diagonal matrices:
This can be extended to the multiplication of any power of :
where the cancellations, along with the fact that the product of diagonal matrices is also diagonal, can speed up the calculation significantly.
Diagonalization is an important and basic method with applications in a number of different scientific areas, such as mathematics, statistical analysis and physics. The practicality can be difficult to absorb directly, but a couple of examples are: finding the interactions between different explanatory variables in statistics and modularizing the energies that are at play in quantum mechanics. In short, mathematics is about the fact that certain square matrices can be divided into components as a product of two matrices, and such that:
where is an orthogonal matrix, is its inverse and is a diagonal matrix. If we look more closely at these two matrices, we see that:
where the columns of are eigenvectors of and the elements of are corresponding eigenvalues of .
A square -matrix is diagonalizable if, and only if, it has linearly independent eigenvectors. An alternative wording is that its eigenspace needs to span the entire .
The matrix is diagonalizable if, and only if, it has n linearly independent eigenvectors
To understand a more technical formulation of requirements for diagonalizability, one needs to know the terms algebraic multiplicity and geometric multiplicity. These two properties are linked to each respective eigenvalue and are defined as:
An eigenvalue's algebraic multiplicity corresponds to its degree of root to the characteristic polynomial .
An eigenvalue's geometric multiplicity corresponds to the dimension of its eigenspace.
The connection between these two terms is that the algebraic multiplicity is always greater than or equal to the geometric multiplicity.
Algebraic multiplicity geometric multiplicity
These two terms dictate the possibility that is diagonalizable. One way of formulating this requirement is that the sum of the geometric multiplicities of its eigenvalues must be equal to . An alternative way of formulating this is that the sum of the algebraic multiplicities of its eigenvalues should be equal to the sum of its geometric multiplicity.
We summarize using the following theorem:
Diagonalizability Let be a square -matrix. Then the following assertions are equivalent:
has linearly independent eigenvectors
has a basis consisting of eigenvectors of
The sum of the geometric multiplicities of the eigenvalues of is equal to
The geometric multiplicity of each eigenvalue of is equal to its algebraic multiplicity
Example of a diagonalizable matrix
To find out if the matrix is diagonalizable, we need to find out if the sum of the geometric multiplicity is 3, since the matrix is . Therefore, we first need to determine the eigenvalues of the matrix, which is done via the equation:
which leads us to the characteristic polynomial:
We can see that we have two eigenvalues, and , respectively. We further see that is a single root and is a double root. Thus, according to the definition of algebraic multiplicity, we have:
has algebraic multiplicity 1
has algebraic multiplicity 2
If we calculate the solution set for each eigenvalue, we have:
From above we see that the dimensions of the eigenvalues are 1 and 2, respectively. We thus state the geometric multiplicities of the eigenvalues:
has geometric multiplicity 1
has geometric multiplicity 2
Now we can conclude that the sum of the algebraic multiplicity of the eigenvalues corresponds to the sum of the geometric multiplicity of the eigenvalues (both are 3). Therefore, the matrix is diagonalizable. To diagonalize we need to determine the matrices and , and to do that we need three linearly independent eigenvectors that can form the basis of each eigenspace. We choose:
and the result of our diagonalization will be:
Example of a non-diagonalizable matrix
Its characteristic polynomial then becomes:
which gives us the opportunity to find the following eigenvalues and their algebraic multiplicities:
with algebraic multiplicity 1
with algebraic multiplicity 2
With these eigenvalues, we produce the respective eigenspaces, analogous to the previous example. It gives us:
Since each eigenspace has dimension one, we can summarize the geometric multiplicities of the eigenvalues:
with geometric multiplicity 1
with geometric multiplicity 1
The matrix is consequently not diagonalizable because the sum of the geometric multiplicities of its eigenvalues is 2, which is less than the sum of the algebraic multiplicities of its eigenvalues, which is 3.