Dodgson condensation explained

In mathematics, Dodgson condensation or method of contractants is a method of computing the determinants of square matrices. It is named for its inventor, Charles Lutwidge Dodgson (better known by his pseudonym, as Lewis Carroll, the popular author), who discovered it in 1866.^[1] The method in the case of an n × n matrix is to construct an (n − 1) × (n − 1) matrix, an (n − 2) × (n − 2), and so on, finishing with a 1 × 1 matrix, which has one entry, the determinant of the original matrix.

General method

This algorithm can be described in the following four steps:

Let A be the given n × n matrix. Arrange A so that no zeros occur in its interior. An explicit definition of interior would be all a_i,j with

i,j\ne1,n

. One can do this using any operation that one could normally perform without changing the value of the determinant, such as adding a multiple of one row to another.

Create an (n − 1) × (n − 1) matrix B, consisting of the determinants of every 2 × 2 submatrix of A. Explicitly, we write

b_i,j=\begin{vmatrix}a_i,&a_i,\ a_i&a_i\end{vmatrix}.

Using this (n − 1) × (n − 1) matrix, perform step 2 to obtain an (n − 2) × (n − 2) matrix C. Divide each term in C by the corresponding term in the interior of A so

c_i,j=\begin{vmatrix}b_i,&b_i,\ b_i&b_i\end{vmatrix}/a_i

Let A = B, and B = C. Repeat step 3 as necessary until the 1 × 1 matrix is found; its only entry is the determinant.

Examples

Without zeros

One wishes to find

\begin{vmatrix} -2&-1&-1&-4\\ -1&-2&-1&-6\\ -1&-1&2&4\\ 2&1&-3&-8 \end{vmatrix}.

All of the interior elements are non-zero, so there is no need to re-arrange the matrix.

We make a matrix of its 2 × 2 submatrices.

\begin{bmatrix} \begin{vmatrix}-2&-1\ -1&-2\end{vmatrix}& \begin{vmatrix}-1&-1\ -2&-1\end{vmatrix}& \begin{vmatrix}-1&-4\ -1&-6\end{vmatrix}\ \\ \begin{vmatrix}-1&-2\ -1&-1\end{vmatrix}& \begin{vmatrix}-2&-1\ -1&2\end{vmatrix}& \begin{vmatrix}-1&-6\ 2&4\end{vmatrix}\ \\ \begin{vmatrix}-1&-1\ 2&1\end{vmatrix}& \begin{vmatrix}-1&2\ 1&-3\end{vmatrix}& \begin{vmatrix}2&4\ -3&-8\end{vmatrix} \end{bmatrix} = \begin{bmatrix} 3&-1&2\\ -1&-5&8\\ 1&1&-4 \end{bmatrix}.

We then find another matrix of determinants:

\begin{bmatrix} \begin{vmatrix}3&-1\ -1&-5\end{vmatrix}& \begin{vmatrix}-1&2\ -5&8\end{vmatrix}\ \\ \begin{vmatrix}-1&-5\ 1&1\end{vmatrix}& \begin{vmatrix}-5&8\ 1&-4\end{vmatrix} \end{bmatrix} = \begin{bmatrix} -16&2\\ 4&12 \end{bmatrix}.

We must then divide each element by the corresponding element of our original matrix. The interior of the original matrix is

\begin{bmatrix} -2&-1\\ -1&2 \end{bmatrix}

, so after dividing we get

\begin{bmatrix} 8&-2\\ -4&6 \end{bmatrix}

.The process must be repeated to arrive at a 1 × 1 matrix.

\begin{bmatrix} \begin{vmatrix} 8&-2\\ -4&6 \end{vmatrix} \end{bmatrix} = \begin{bmatrix} 40 \end{bmatrix}.

Dividing by the interior of the 3 × 3 matrix, which is just −5, gives

\begin{bmatrix}-8\end{bmatrix}

and −8 is indeed the determinant of the original matrix.

With zeros

Simply writing out the matrices:

\begin{bmatrix} 2&-1&2&1&-3\\ 1&2&1&-1&2\\ 1&-1&-2&-1&-1\\ 2&1&-1&-2&-1\\ 1&-2&-1&-1&2 \end{bmatrix} \to \begin{bmatrix} 5&-5&-3&-1\\ -3&-3&-3&3\\ 3&3&3&-1\\ -5&-3&-1&-5 \end{bmatrix} \to \begin{bmatrix} -15&6&12\\ 0&0&6\\ 6&-6&8 \end{bmatrix}.

Here we run into trouble. If we continue the process, we will eventually be dividing by 0. We can perform four row exchanges on the initial matrix to preserve the determinant and repeat the process, with most of the determinants precalculated:

\begin{bmatrix} 1&2&1&-1&2\\ 1&-1&-2&-1&-1\\ 2&1&-1&-2&-1\\ 1&-2&-1&-1&2\\ 2&-1&2&1&-3 \end{bmatrix} \to \begin{bmatrix} -3&-3&-3&3\\ 3&3&3&-1\\ -5&-3&-1&-5\\ 3&-5&1&1 \end{bmatrix} \to \begin{bmatrix} 0&0&6\\ 6&-6&8\\ -17&8&-4 \end{bmatrix} \to \begin{bmatrix} 0&12\\ 18&40 \end{bmatrix} \to \begin{bmatrix} 36 \end{bmatrix}.

Hence, we arrive at a determinant of 36.

Desnanot–Jacobi identity and proof of correctness of the condensation algorithm

The proof that the condensation method computes the determinant of the matrix if no divisions by zero are encountered is based on an identity known as the Desnanot–Jacobi identity (1841) or, more generally, the Sylvester determinant identity (1851).^[2]

Let

M=(m_i,j

	k
)
	i,j=1

be a square matrix, and for each

1\lei,j\lek

, denote by

	j
M
	i

the matrix that results from

by deleting the

-th row and the

-th column. Similarly, for

1\lei,j,p,q\lek

, denote by

	p,q
M
	i,j

the matrix that results from

by deleting the

-th and

-th rows and the

-th and

-th columns.

Desnanot–Jacobi identity

\det(M)

	1,k
\det(M
	1,k

	k)
\det(M
	k

	k)
\det(M
	1

	1).
\det(M
	k

Proof of the correctness of Dodgson condensation

Rewrite the identity as

\det(M)=

\det(M

	k)
\det(M
	1

	1)
\det(M
	k

	1,k
\det(M		)
	1,k

Now note that by induction it follows that when applying the Dodgson condensation procedure to a square matrix

of order

, the matrix in the

-th stage of the computation (where the first stage

k=1

corresponds to the matrix

itself) consists of all the connected minors of order

, where a connected minor is the determinant of a connected

k x k

sub-block of adjacent entries of

. In particular, in the last stage

k=n

, one gets a matrix containing a single element equal to the unique connected minor of order

, namely the determinant of

Proof of the Desnanot-Jacobi identity

We follow the treatment in the book Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture;^[3] an alternative combinatorial proof was given in a paper by Doron Zeilberger.^[4]

Denote

a_i,j=(-1)^i+j

	j)
\det(M
	i

(up to sign, the

(i,j)

-th minor of

), and define a

k x k

matrix

M'=\begin{pmatrix}a_1,1&0&0&0&\ldots&0&a_k,1\\ a_1,2&1&0&0&\ldots&0&a_k,2\\ a_1,3&0&1&0&\ldots&0&a_k,3\\ a_1,4&0&0&1&\ldots&0&a_k,4\\ \vdots&\vdots&\vdots&\vdots&&\vdots&\vdots\\ a_1,k-1&0&0&0&\ldots&1&a_k,k-1\\ a_1,k&0&0&0&\ldots&0&a_k,k\end{pmatrix}.

(Note that the first and last column of

are equal to those of the adjugate matrix of

). The identity is now obtained by computing

\det(MM')

in two ways. First, we can directly compute the matrix product

MM'

(using simple properties of the adjugate matrix, or alternatively using the formula for the expansion of a matrix determinant in terms of a row or a column)to arrive at

MM'=\begin{pmatrix} \det(M)&m_1,2&m_1,3&\ldots&m_1,k-1&0\\ 0&m_2,2&m_2,3&\ldots&m_2,k-1&0\\ 0&m_3,2&m_3,3&\ldots&m_3,k-1&0\\ \vdots&\vdots&\vdots&&\vdots&\vdots&\vdots\\ 0&m_k-1,2&m_k-1,3&\ldots&m_k-1,k-1&0\\ 0&m_k,2&m_k,3&\ldots&m_k,k-1&\det(M) \end{pmatrix}

where we use

m_i,j

to denote the

(i,j)

-th entry of

. The determinant of this matrix is

\det(M)² ⋅

	1,k
\det(M
	1,k

)

.
Second, this is equal to the product of the determinants,

\det(M) ⋅ \det(M')

. But clearly

\det(M')=a_1,1a_k,k-a_k,1a_1,k=

	k)
\det(M
	k

	k)
\det(M
	1

	1),
\det(M
	k

so the identity follows from equating the two expressions we obtained for

\det(MM')

and dividing out by

\det(M)

(this is allowed if one thinks of the identities as polynomial identities over the ring of polynomials in the

k²

indeterminate variables

(m_i,j

	k
)
	i,j=1

Notes and References

Dodgson, C. L. . Condensation of Determinants, Being a New and Brief Method for Computing their Arithmetical Values . Proceedings of the Royal Society of London . 15 . 1866–1867 . 150–155 . 1866RSPS...15..150D .
Sylvester . James Joseph . On the relation between the minor determinants of linearly equivalent quadratic functions . Philosophical Magazine . 1 . 1851 . 295–305.
Cited in Akritas . A. G. . Akritas . E. K. . Malaschonok . G. I. . 10.1016/S0378-4754(96)00035-3 . Various proofs of Sylvester's (determinant) identity . Mathematics and Computers in Simulation . 42 . 4–6 . 585 . 1996 .
Book: Bressoud, David. 1999. Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture . Cambridge University Press. 9781316582756.
Zeilberger. Doron. 1997. Dodgson's Determinant-Evaluation Rule Proved by Two-Timing Men and Women . Electron. J. Comb. . 4. 2. article R22 . 10.37236/1337. October 27, 2023. free.