Sylvester Method

 

 

Sylvester's Formula

Suppose that A is a diagonalizable \( n \times n \) matrix; this means that all its eigenvalues are not defective and there exists a basis of n linearly independent eigenvectors. Now assume that its minimal polynomial (the polynomial of least possible degree that anihilates the matrix) is a product of simple terms:

\[ \psi (\lambda ) = \left( \lambda - \lambda_1 \right) \left( \lambda - \lambda_2 \right) \cdots \left( \lambda - \lambda_k \right) , \]

where \( \lambda_1 , \lambda_2 , \ldots , \lambda_k \) are distinct eigenvalues of the matrix A ( \( k \le n \) ).

Let \( f(\lambda ) \) be a function defined on the spectrum of the matrix A. The last condition means that every eigenvalue λ_i is in the domain of f, and that every eigenvalue λ_i with multiplicity m_i > 1 is in the interior of the domain, with f being (m_i — 1) times differentiable at λ_i. We build a function \( f\left( {\bf A} \right) \) of diagonalizable square matrix A according to James Sylvester, who was an English lawyer and music tutor before his appointment as a professor of mathematics in 1885. To define a function of a square matrix, we need to construct k Sylvester auxiliary matrices for each distinct eigenvalue \( \lambda_i , \quad i= 1,2,\ldots k: \)

\[ {\bf Z}_{\lambda_i} = {\bf Z}_{i} = \dfrac{\left( {\bf A} - \lambda_1 {\bf I} \right) \cdots \left( {\bf A} - \lambda_{i-1} {\bf I} \right) \left( {\bf A} - \lambda_{i+1} {\bf I} \right) \cdots \left( {\bf A} - \lambda_k {\bf I} \right)}{(\lambda_i - \lambda_1 ) \cdots (\lambda_i - \lambda_{i-1} ) (\lambda_i - \lambda_{i+1} ) \cdots (\lambda_i - \lambda_k )} . \]

Now we define the function \( f\left( {\bf A} \right) \) according to the formula:

\[ f\left( {\bf A} \right) = \sum_{i=1}^k f(\lambda_i )\,{\bf Z}_i . \]

Each Sylvester's matrix is a projection matrix on eigenspace of the corresponding eigenvalue.

Example. Consider a 2-by-2 matrix

\[ {\bf A} = \begin{bmatrix} 4&5 \\ -1& -2 \end{bmatrix} \]

that has two distinct eigenvalues \( \lambda_1 =3 \) and \( \lambda_2 =-1 . \) The eigenvectors corresponding these eigenvalues are

\[ {\bf v}_1 = \left\langle -5 , \ 1 \right\rangle^T \qquad {\bf v}_2 = \left\langle 1 , \ -1 \right\rangle^T , \]

respectively. Since the minimal polynomail is \( \psi (\lambda ) = (\lambda -3)(\lambda +1) \) is a product of simple factors, we build Sylvester's auxiliary matrices:

\[ {\bf Z}_3 = \frac{{\bf A} +1}{3+1} = \frac{1}{4} \begin{bmatrix} 5&5 \\ -1& -1 \end{bmatrix} \quad\mbox{and} \quad {\bf Z}_{-1} = \frac{{\bf A} -3}{-1-3} = \frac{1}{4} \begin{bmatrix} -1&-5 \\ 1&5 \end{bmatrix} . \]

These matrices are projectors on eigenspaces:

\[ {\bf Z}_3^2 = {\bf Z}_3 \qquad \mbox{and} \qquad {\bf Z}_{-1}^2 = {\bf Z}_{-1} . \]

Let us take an arbitrary vector, say \( {\bf x} = \langle 2, \ 2 \rangle^T , \) and expand it with respect to eigenvectors:

\[ {\bf x} = \begin{bmatrix} 2 \\ 2 \end{bmatrix} = a\, {\bf v}_1 + b\,{\bf v}_2 = a \begin{bmatrix} -5 \\ 1 \end{bmatrix} + b \begin{bmatrix} 1 \\ -1 \end{bmatrix} , \]

where coefficients a and b need to be determined. Of course, we can find them by solving the system of linear equations:

\[ \begin{split} 2 &= a(-5) + b , \\ 2 &= a - b . \end{split} \]

However, the easiet way is to apply Sylvester's auxiliary matrices:

\begin{align*} {\bf Z}_3 \, {\bf x} &= \frac{1}{4} \begin{bmatrix} 5&5 \\ -1& -1 \end{bmatrix} \, \begin{bmatrix} 2\\ 2 \end{bmatrix} = \frac{1}{2} \begin{bmatrix} 5&5 \\ -1& -1 \end{bmatrix} \, \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \frac{1}{2} \begin{bmatrix} 10 \\ 2 \end{bmatrix} = \begin{bmatrix} 5\\ -1 \end{bmatrix} = -{\bf v}_1 , \\ {\bf Z}_{-1}\, {\bf x} &= \frac{1}{4} \begin{bmatrix} -1&-5 \\ 1&5 \end{bmatrix} \, \begin{bmatrix} 2\\ 2 \end{bmatrix} = \begin{bmatrix} -3 \\ 3 \end{bmatrix} = -3\, {\bf v}_2 . \end{align*}

Therefore, coefficients are \( a =-1 \) and \( b =-3 . \) So we get the expansion:

\[ {\bf x} = \begin{bmatrix} 2 \\ 2 \end{bmatrix} = - {\bf v}_1 - 3\,{\bf v}_2 = - \begin{bmatrix} -5 \\ 1 \end{bmatrix} -3 \begin{bmatrix} 1 \\ -1 \end{bmatrix} . \]

We can define some functions, for instance, \( f(\lambda ) = e^{\lambda \,t} \) and \( g(\lambda ) = \cos \left( \lambda \,t\right) . \) Indeed,

\begin{align*} f({\bf A}) &= e^{3t}\, {\bf Z}_3 + e^{-t}\, {\bf Z}_{-1} = e^{3t}\, \frac{1}{4} \begin{bmatrix} 5&5 \\ -1& -1 \end{bmatrix} + e^{-t} \, \frac{1}{4} \begin{bmatrix} -1&-5 \\ 1&5 \end{bmatrix} , \\ g({\bf A}) &= \cos (3t) \, {\bf Z}_3 + \cos\, t\, {\bf Z}_{-1} = \frac{\cos \,3t}{4} \begin{bmatrix} 5&5 \\ -1& -1 \end{bmatrix} + \frac{\cos\, t}{4} \begin{bmatrix} -1&-5 \\ 1&5 \end{bmatrix} .\end{align*}

sage: M.rank()
sage: M.right_nullity()

Example 1.7.1: Consider the \( 3 \times 3 \) matrix \( {\bf A} = \begin{bmatrix} 1&4&16 \\ 18&20&4 \\ -12&-14&-7 \end{bmatrix} \) that has three distinct eigenvalues

A = {{1,4,16},{18,20,4},{-12,-14,-7}}

Eigenvalues[A]

Out[2]= 9, 4, 1

To determine a function of matrix, \( f ({\bf A} ) , \) we first find Sylvester's auxiliary matrices:

sage: M.rank()
sage: M.right_nullity()

 

Example 1.7.2: Consider the \( 3 \times 3 \) matrix \( {\bf A} = \begin{bmatrix} -20&-42&-21 \\ 6&13&6 \\ 12&24&13 \end{bmatrix} \) that has two distinct eigenvalues

Eigenvalues[A]

Out[2]= 4, 1, 1

Then we define the resolvent:

resolvent = FullSimplify[Inverse[lambda*IdentityMatrix[3] - A]]

Out[2]= {{(-25 + lambda)/( 4 - 5 lambda + lambda^2), -(42/(4 - 5 lambda + lambda^2)), -(21/( 4 - 5 lambda + lambda^2))}, {6/(4 - 5 lambda + lambda^2), ( 8 + lambda)/(4 - 5 lambda + lambda^2), 6/( 4 - 5 lambda + lambda^2)}, {12/(4 - 5 lambda + lambda^2), 24/( 4 - 5 lambda + lambda^2), (8 + lambda)/(4 - 5 lambda + lambda^2)}}

Using the resolvent, we define Sylvester's auxiliary matrices (that are actiually projector matrices on eigenspace)

Z1 = FullSimplify[(lambda - 1)*resolvent] /. lambda -> 1

Out[3]= {{8, 14, 7}, {-2, -3, -2}, {-4, -8, -3}}

Z4 = FullSimplify[(lambda - 4)*resolvent] /. lambda -> 4

Out[4]= {{-7, -14, -7}, {2, 4, 2}, {4, 8, 4}}

sage: M.rank()
sage: M.right_nullity()

 

Example 1.7.3:

sage: M.rank()
sage: M.right_nullity()