Matrix Operations in kdb+/q

In q, a matrix (an array of m x n numbers) is represented as a list of lists. For example, here is a matrix with 2 rows and 3 columns:

q)A:(1 2 3;4 5 6)
q)A
1 2 3
4 5 6

Matrix Addition and Subtraction
If A and B are matrices of the same size, then they can be added and subtracted. To find the entries of A + B, you simply add the corresponding entries of A and B. To find A - B, subtract corresponding entries. If A and B have different sizes, you will get a 'length error.

q)A:(1 2;3 4)
q)B:(5 6;7 8)
q)A+B
6  8
10 12
q)B-A
4 4
4 4
q)C:(1 1 1;2 2 2)
q)A+C
'length
  [0]  A+C
        ^

Scalar Multiplication
If A is a matrix and k is a scalar, then the matrix kA is obtained by multiplying each entry of A by k.

q)A:(1 2;3 4)
q)k:2
q)k*A
2 4
6 8

Matrix Multiplication
First, in order to multiply two matrices A and B, the number of columns of A must match the number of rows of B. If A is m x n and B is n x p, then the size of the product matrix AB will be m x p. In order to calculate AB, you have to take the dot product (multiply corresponding numbers and then add them up) of each row vector in A and the corresponding column vector in B. This can be done in q using the mmu (or $) operator.

q)A:(1 2f;3 4f)
q)B:(5 6f;7 8f)
q)A
1 2
3 4
q)B
5 6
7 8
q)A mmu B
19 22
43 50

Identity Matrix
This is a square matrix with 1s on the diagonal and 0s everywhere else.

q)I:{`float${x=/:x}til x}
q)I 3
1 0 0
0 1 0
0 0 1

Matrix Inverse
The inverse of A is A^-1 if AA^-1=A^-1A=I, where I is the identity matrix. Use the inv function to find the inverse of a matrix.

q)A:(1 2f;3 4f)
q)inv A
-2  1
1.5 -0.5
q)A mmu inv A
1 1.110223e-016
0 1

Matrix Tranpose
The tranpose A^T of a matrix A is a flipped version of the original matrix which is obtained by changing its rows into columns (or equivalently, its columns into rows). This can be done by using the flip operation in q.

q)A:(1 2 3;4 5 6)
q)flip A
1 4
2 5
3 6

Upper Triangular Matrix
This is a square matrix in which all the entries below the main diagonal are zero.

q)upperTriangle:{`float${x<=\:x}til x}
q)upperTriangle 3
1 1 1
0 1 1
0 0 1

Lower Triangular Matrix
This is a square matrix in which all the entries above the main diagonal are zero.

q)lowerTriangle:{`float${x>=\:x}til x}
q)lowerTriangle 3
1 0 0
1 1 0
1 1 1