Sylvester's formula

In matrix theory, Sylvester's formula or Sylvester's matrix theorem (named after J. J. Sylvester) or Lagrange−Sylvester interpolation expresses an analytic function f (A) of a matrix A as a polynomial in A, in terms of the eigenvalues and eigenvectors of A. It states that

f

(

A

)

=

∑

i

=

1

k

f

(

λ

i

)

A

i

,

{\displaystyle f(A)=\sum _{i=1}^{k}f(\lambda _{i})~A_{i}~,}

where the λi are the eigenvalues of A, and the matrices

A

i

≡

∏

j

=

1

j

≠

i

k

1

λ

i

−

λ

j

(

A

−

λ

j

I

)

{\displaystyle A_{i}\equiv \prod _{j=1 \atop j\neq i}^{k}{\frac {1}{\lambda _{i}-\lambda _{j}}}\left(A-\lambda _{j}I\right)}

are the corresponding Frobenius covariants of A, which are (projection) matrix Lagrange polynomials of A.

Conditions

Sylvester's formula applies for any diagonalizable matrix A with k distinct eigenvalues, λ1, ..., λk, and any function f defined on some subset of the complex numbers such that f(A) is well defined. The last condition means that every eigenvalue λi is in the domain of f, and that every eigenvalue λi with multiplicity mi > 1 is in the interior of the domain, with f being (mi - 1) times differentiable at λi.

Example

Consider the two-by-two matrix:

A

=

[

1

3

4

2

]

.

{\displaystyle A={\begin{bmatrix}1&3\\4&2\end{bmatrix}}.}

This matrix has two eigenvalues, 5 and −2. Its Frobenius covariants are

A

1

=

c

1

r

1

=

[

3

4

]

[

1

7

1

7

]

=

[

3

7

3

7

4

7

4

7

]

=

A

+

2

I

5

−

(

−

2

)

A

2

=

c

2

r

2

=

[

1

7

−

1

7

]

[

4

−

3

]

=

[

4

7

−

3

7

−

4

7

3

7

]

=

A

−

5

I

−

2

−

5

.

{\displaystyle {\begin{aligned}A_{1}&=c_{1}r_{1}={\begin{bmatrix}3\\4\end{bmatrix}}{\begin{bmatrix}{\frac {1}{7}}&{\frac {1}{7}}\end{bmatrix}}={\begin{bmatrix}{\frac {3}{7}}&{\frac {3}{7}}\\{\frac {4}{7}}&{\frac {4}{7}}\end{bmatrix}}={\frac {A+2I}{5-(-2)}}\\A_{2}&=c_{2}r_{2}={\begin{bmatrix}{\frac {1}{7}}\\-{\frac {1}{7}}\end{bmatrix}}{\begin{bmatrix}4&-3\end{bmatrix}}={\begin{bmatrix}{\frac {4}{7}}&-{\frac {3}{7}}\\-{\frac {4}{7}}&{\frac {3}{7}}\end{bmatrix}}={\frac {A-5I}{-2-5}}.\end{aligned}}}

Sylvester's formula then amounts to

f

(

A

)

=

f

(

5

)

A

1

+

f

(

−

2

)

A

2

.

{\displaystyle f(A)=f(5)A_{1}+f(-2)A_{2}.\,}

For instance, if f is defined by f(x) = x−1, then Sylvester's formula expresses the matrix inverse f(A) = A−1 as

1

5

[

3

7

3

7

4

7

4

7

]

−

1

2

[

4

7

−

3

7

−

4

7

3

7

]

=

[

−

0.2

0.3

0.4

−

0.1

]

.

{\displaystyle {\frac {1}{5}}{\begin{bmatrix}{\frac {3}{7}}&{\frac {3}{7}}\\{\frac {4}{7}}&{\frac {4}{7}}\end{bmatrix}}-{\frac {1}{2}}{\begin{bmatrix}{\frac {4}{7}}&-{\frac {3}{7}}\\-{\frac {4}{7}}&{\frac {3}{7}}\end{bmatrix}}={\begin{bmatrix}-0.2&0.3\\0.4&-0.1\end{bmatrix}}.}

Generalization

Sylvester's formula is only valid for matrices with no repeated eigenvalues; an extension due to Arthur Buchheim, based on Hermite interpolating polynomials, covers the general case:

f

(

A

)

=

∑

i

=

1

s

[

∑

j

=

0

n

i

−

1

1

j

!

ϕ

i

(

j

)

(

λ

i

)

(

A

−

λ

i

I

)

j

∏

j

=

1

,

j

≠

i

s

(

A

−

λ

j

I

)

n

j

]

{\displaystyle f(A)=\sum _{i=1}^{s}\left[\sum _{j=0}^{n_{i}-1}{\frac {1}{j!}}\phi _{i}^{(j)}(\lambda _{i})\left(A-\lambda _{i}I\right)^{j}\prod _{j=1,j\neq i}^{s}\left(A-\lambda _{j}I\right)^{n_{j}}\right]}

,

where

ϕ

i

(

t

)

:=

f

(

t

)

/

∏

j

≠

i

(

t

−

λ

j

)

n

j

{\displaystyle \phi _{i}(t):=f(t)/\prod _{j\neq i}\left(t-\lambda _{j}\right)^{n_{j}}}

.

A concise form is further given by Hans Schwerdtfeger,

f

(

A

)

=

∑

i

=

1

s

A

i

∑

j

=

0

n

i

−

1

f

(

j

)

(

λ

i

)

j

!