atrices are very useful mathematical objects that science and technology professionals use to describe real-life scenarios and build abstract models for those scenarios. You will find matrices used in economics and statistics, in cryptography and genetics, and in computer graphics. Unfortunately for database developers, SQL and SQL Server do not support direct operations on matrices. However, because tables and matrices share the same structures, SQL allows easy manipulations with matrices. This article demonstrates a few SQL techniques for performing some basic operations on matrices.

Some Matrix Operation Definitions
A matrix is a rectangular array of elements, where the elements can be symbols, symbolic expressions, or numbers (see Figure 1 below):

In general, matrix A in Figure 1 can be denoted by the following:

In this denotation, m is the number of rows and n is the number of columns in the matrix A. Notation m x n represents the size of the matrix, so matrix A can also be denoted by A_{m x n} or by [A]_{m x n}.

The first matrix in Figure 1 has three rows and four columns. Therefore, the size of that matrix is 3 x 4. The size of the second matrix in Figure 1 is 3 x 3. This matrix is also called a square matrix, because the number of rows is equal to the number of columns.

Each element of the matrix A_{m x n} is denoted by a_ij, where the row number is 1 ? i ? m and the column number is 1 ? j ? n. The element a₂₃ of the first matrix is located in the third column of the second row. Similarly, you can find the location of element a₂₃ in the second matrix. The value of that element is a₂₃ = 20.

Simple Operations on Matrices
Here are some simple operations you can implement on matrices:

1) Matrix Addition
Matrices A and B can be added, if they are the same size (e.g., m x n). The matrix sum C = A + B also will be a m x n matrix, where for each element c_ij = a_ij + b_ij and 1 ? i ? m and 1 ? j ? n.

To add these two matrices:

You would use this solution:

Matrix addition is commutative (e.g., A + B = B + A) and associative (e.g., C + (D + E) = (C + D) + E).

2) Matrix Subtraction
Two matrices (A and B) can be subtracted if they are the same size (e.g., k x l). The matrix difference D = A ? B will also be a k x l matrix with the element d_ij = a_ij ? b_ij, where 1 ? i ? k and 1 ? j ? l.

To subtract matrix B from matrix A:

You would use this solution:

Matrix subtraction is neither commutative nor associative.

3) Matrix Scalar Multiplication
If A is an m x n matrix and the constant k is a number, then the scalar product of k and A is a new matrix B = kA, where b_ij = ka_ij and 1 ? i ? m and 1 ? j ? n.

To find B = 5?A, where:

You would use this solution:

Scalar multiplication is commutative and associative:

k(AB) = (kA)B = A(kB) = (AB)k

4) Matrices Multiplication
Two matrices (A and B) can be multiplied if they are compatible, meaning that the first (left) matrix has as many columns as the second (right) matrix has rows. If matrices A and B meet this condition, you can multiply them. The result of such a multiplication will be matrix C, which has as many rows as matrix A and as many columns as matrix B:

Amxn ?Bnxp  = Cmxp

The rules for matrices multiplication are not as trivial as for scalar matrix multiplication. To calculate the ij-element of the product matrix, you need to multiply the i^th row vector of the first matrix by the j^th column vector of the second matrix. The multiplication of a row vector by a column vector goes as follows:

1. Multiply the first elements of the row and column vectors to get product 1.
2. Multiply the second elements of the row and column vectors to get product 2.
. . . . . . . . . . . . .
N. Multiply the Ns elements of the row and column vectors to get product N.
Lastly, sum up all the products.

The result of such a calculation will be a scalar.

Figure 2 shows the algorithm in math notation for A_mxn ?B_nxp = C_mxp (where i = 1, 2, . . m, and j =1, 2, . . p.).

To find product C = A ? B, where:

You would use this solution:

Matrices multiplication is not commutative (i.e., A?B ? B?A). Indeed, in Example 4, the multiplication A_3×2 ? B_2×4 is allowed, because matrices A_3×2 and B_2×4 are compatible. However, the multiplication B_2×4 ? A_3×2 would be illegal, because the number of columns in the left matrix is not equal to the number of rows in the right matrix. But even if both A?B and B?A are compatible, that doesn?t guarantee the same result.

Consider two square matrices:

However, matrices multiplication is associative and distributive:

• associative – A?(B?C) = (A?B)?C
• distributive – A?(B + C) = A?B + A?C
• distributive – (A + B)C = A?C + B?C

5) Matrix Transpose
The transposing of a matrix simply means an exchange between the rows and the columns. Suppose you have an m x n matrix, A_mxn. If you transpose that matrix, you will get an n x m matrix, B_nxm, with the following correspondence between the elements of A and B:

aixj   = bjxi, for all i and j.

The most common notation for the transposing of matrix A is A^T.

Find the transpose of a matrix A as follows:

Matrices Addition/Subtraction Implementation in SQL
To demonstrate the implementation of matrices addition and subtraction in SQL, create tables A and B, which will represent two sample matrices A_5×5 and B_5×5 (see Listing 1).

Since matrices A and B are the same size, you can add and subtract them. The implementation in SQL is very straightforward (see Listing 2):

Matrices Multiplication Implementation in SQL
Since matrix scalar multiplication in SQL is obvious?simply multiply all columns in the table by a constant?it won’t be discussed here. Instead, let’s explore a more complicated operation: matrices multiplication. The example in this section will implement matrices multiplication in three steps:

1. Transform matrices A and B into two-dimensional arrays.
2. Calculate the product.
3. Transform calculated product back into matrix form.

Before you start, rebuild tables A and B. This time, they will represent two sample matrices, A_7×8 and B_8×9 (see Listing 3).

Matrices A_7×8 and B_8×9 are compatible because the number of columns in matrix A_7×8 is equal to the number of rows in matrix B_8×9. The result of multiplication is that matrix-product C_7×9 will have seven rows and nine columns: A_7×8 ? B_8×9 = C_7×9.

The code in the Listing 4 demonstrates how to perform matrices multiplication. All three steps described earlier in this section are implemented in the one query in Listing 4.

The first and second CTE queries transform matrices A and B into two-dimensional arrays. The third CTE query calculates matrices product according to the formula in Figure 2. The last SQL statement transforms a two-dimensional array into the matrix result.

Using the same approach with two-dimensional arrays, you can easily perform addition and subtraction.

Matrix Transpose Implementation in SQL
To transpose matrix A, just transform that matrix into a two-dimensional array and then transform it back into the matrix form. For reverse transformation, however, you replace the rows with the columns and the columns with the rows (see Listing 5).

Now you know a few SQL techniques for performing addition, subtraction, multiplication, and transpose on matrices.