The Income Statement as a Matrix

 

In the matrix version of the Balance Sheet, shown below and in the previous post, all of the credits cancel out all of the debits, bringing the balance to zero, as shown by the matrix elements in the lower right hand corner. This, of course indicates a balance of the double-entry bookkeeping.

 

We can make another matrix based on the same data, but only showing a subset of the accounts (the accounts in the matrix are shown as A, B, C, D, and E to keep the demonstration at an abstraction level). The accounts that we would include are the Revenue accounts and the Expense Accounts. And the matrix would represent a linear algebra view of an Income Statement.” 

 

Accounts	A Credits	B Credits	C Credits	D Credits	E Credits	Debit Totals
A Debits	0	to A from B	to A from C	to A from D	to A from E	A Debit Total
B Debits	to B from A	0	to B from C	to B from D	to B from E	B Debit Total
C Debits	to C from A	to C from B	0	to C from D	to C from E	C Debit Total
D Debits	to D from A	to D from B	to D from C	0	to D from E	D Debit Total
E Debits	to E from A	to E from B	to E from C	to E from D	0	E Debit Total
Credit Totals	A Credit Total	B Credit Total	C Credit Total	D Credit Total	E Credit Total	0

 

Since only a subset of the accounts are shown, the summary that we find in a Balance Sheet would very likely no longer show a balance of zero. The matrix element in the lower right corner would now show a debit balance, indicating a profitable period, or a credit balance showing net losses. The debits to the Revenue accounts would offset and exceed the credits shown in the expense accounts where a profitable Income Statement is given. Otherwise, the credits would be in excess.

 

Again this matrix Income Statement is much easier to read and contains an order of magnitude more information. The future of financial analysis belongs to matrix arithmetic and linear algebra.

Linear Algebra as a Balance Sheet

 

The matrix below is the accounting balance sheet of the future. It is a two-dimensional table like those found in the linear algebra branch of mathematics.. Each element of the matrix is a vector from an account that is credited to an account that is debited (the use of capital letters is preferred for this abstract demonstration over the more concrete names such as “Cash” or “Accumulated Depreciation”). Each vector is consistent with the use of a credited account as the origin of the transferred resource and the a debited account as its destination. This modern balance sheet contains all the information found in the traditional balance sheet, but it also contains an order of magnitude more information. This matrix of vectors is explained further in my book, “The Tao of Financial Information.” 

 

Accounts	A Credits	B Credits	C Credits	D Credits	E Credits	Debit Totals
A Debits	0	to A from B	to A from C	to A from D	to A from E	A Debit Total
B Debits	to B from A	0	to B from C	to B from D	to B from E	B Debit Total
C Debits	to C from A	to C from B	0	to C from D	to C from E	C Debit Total
D Debits	to D from A	to D from B	to D from C	0	to D from E	D Debit Total
E Debits	to E from A	to E from B	to E from C	to E from D	0	E Debit Total
Credit Totals	A Credit Total	B Credit Total	C Credit Total	D Credit Total	E Credit Total	0

 

The current conventional balance sheet is very primitive and is based on the traditions and mathematics of the historical medieval period, a time when western civilization was first using Arabic numbers and did not believe that negative numbers existed. The accounting analysis of the future can be found in the practice of linear algebra.