SQL Reference

Matrix Functions and Operators

Matrix Functions
Matrixes store values in DOUBLE format.
_R{} and _C{} are vector notation. For example, _R{ 1,2,3 } is a row vector with values 1, 2, and 3.
Function
Syntax
Purpose
Example
Result
Constructor
make_matrix_ixj(i, j, e_00, …​, e_ij)
Creates an ixj matrix with elements e_00, …​, e_ij
make_matrix_1x3(1,3,1,2,3)
{ {1.0,2.0,3.0} }
Constructor (2)
matrix_from_text(string)
Creates a matrix from the specified string
matrix_from_text('{ {1,2,3}, {1,2,3} }')
{ {1.0,2.0,3.0}, {1.0,2.0,3.0} }
+ (matrix addition)
matrix + matrix
Performs matrix addition
{ {1,2,3}, {4,5,6} } + { {4,5,6}, {1,2,3} }
{ {5.0,7.0,9.0}, {5.0,7.0,9.0} }
- (matrix subtraction)
matrix - matrix
Performs matrix subtraction
{ {1,2,3}, {4,5,6} } - { {4,5,6}, {1,2,3} }
{ {-3.0,-3.0,-3.0}, {3.0,3.0,3.0} }
* (scalar multiplication)
numeric * matrix
Performs scalar multiplication
2 * { {4,5,6}, {1,2,3} }
{ {2.0,4.0,6.0}, {8.0,10.0,12.0} }
* (matrix multiplication)
matrix * matrix
Performs matrix multiplication
{ {1,2,3}, {4,5,6} } * { {1,2}, {3,4}, {5,6} }
{ {22.0,28.0}, {49.0,64.0} }
/ (scalar division)
matrix / numeric
Performs scalar division
{ {1,2,3}, {4,5,6} } / 2
{ {0.5,1.0,1.5}, {2.0,2.5,3.0} }
transpose
transpose(matrix)
Returns transpose of the matrix
transpose( { {1,2,3}, {4,5,6} } )
{ {1.0,4.0}, {2.0,5.0}, {3.0,6.0} }
determinant
det(matrix)
Returns determinant of the matrix as a double
det( { {1,2}, {3,4} } )
-2.0
inverse
inverse(matrix)
Returns inverse of a square, invertible matrix
inverse( { {1,2}, {3,4} } )
{ {-2.0, 1.0}, {1.5, -0.5} }
trace
matrix_trace(matrix)
Returns trace of a square matrix as a double
matrix_trace( { {1,2,3}, {4,5,6}, {7,8,9} } )
15.0
vector sum
vector_sum(matrix)
Returns sum of elements in a 1D matrix/vector
vector_sum( _R{ 1,2,3 } )
vector_sum( _C{ 1,2,3 } )
6.0
vector min
vector_min(matrix)
Returns minimum of elements in a 1D matrix/vector
vector_min( _R{ 1,2,3 } )
vector_min( _C{ 1,2,3 } )
1.0
vector max
vector_max(matrix)
Returns maximum of elements in a 1D matrix/vector
vector_max( _R{ 1,2,3 } )
vector_max( _C{ 1,2,3 } )
3.0
vector argmin
vector_argmin(matrix)
Returns the argmin of a 1D matrix or vector. The index starts at 0.
vector_argmin(_R{1, -1, 5, 0})
1
vector argmax
vector_argmax(matrix)
Returns the argmax of a 1D matrix or vector. The index starts at 0.
vector_argmax(_C{1, -1, 5, 0})
2
softmax
softmax(matrix)
Returns the softmax of a 1D matrix or vector.
softmax(_R{1, 1})
{{0.5, 0.5}}
cross entropy loss
cross_entropy_loss(matrix, matrix)
Returns the cross entropy loss of two one-dimensional matrixes or vectors.
cross_entropy_loss( _R{ 0.2, 0.5, 0.8 }, _C{ 0,1,0 } )
﻿
cross_entropy_loss( _C{ 0.2, 0.5, 0.8}, _R{ 0,1,0 } )
﻿
cross_entropy_loss( _R{ 0.2, 0.5, 0.8 }, _R{ 0,1,0 } )
﻿
cross_entropy_loss( _C{ 0.2, 0.5, 0.8 }, _C{ 0,1,0} )
0.69
log loss 
log_loss(matrix, matrix)
Returns the log loss of two one-dimensional matrixes or vectors.
log_loss( _R{ 0.2, 0.5, 0.8 }, _C{ 0,1,0 } )

log_loss( _C{ 0.2, 0.5, 0.8}, _R{ 0,1,0 } )

log_loss( _R{ 0.2, 0.5, 0.8 }, _R{ 0,1,0 } )

log_loss( _C{ 0.2, 0.5, 0.8 }, _C{ 0,1,0} )
2.53
logits loss 
logits_loss(matrix, matrix)
Returns the logits loss of two one-dimensional matrixes or vectors.
logits_loss( _R{ 0.2, 0.5, 0.8 }, _C{ 0,1,0 } )
logits_loss( _C{ 0.2, 0.5, 0.8}, _R{ 0,1,0 } )
logits_loss( _R{ 0.2, 0.5, 0.8 }, _R{ 0,1,0 } )
logits_loss( _C{ 0.2, 0.5, 0.8 }, _C{ 0,1,0} )
2.44
hinge loss 
hinge_loss(matrix, matrix)
Returns the hinge loss of two one-dimensional matrixes or vectors.
hinge_loss( _R{ 0.2, 0.5, 0.8 }, _C{ 0,1,0 } )
hinge_loss( _C{ 0.2, 0.5, 0.8}, _R{ 0,1,0 } )
hinge_loss( _R{ 0.2, 0.5, 0.8 }, _R{ 0,1,0 } )
hinge_loss( _C{ 0.2, 0.5, 0.8 }, _C{ 0,1,0} )
3.5
dot product
dot(matrix, matrix)
Returns dot product of two one-dimensional matrixes/vectors
dot( _R{ 1,2,3 }, _C{ 1,2,3 } )
dot( _C{ 1,2,3 }, _R{ 1,2,3 } )
dot( _R{ 1,2,3 }, _R{ 1,2,3 } )
dot( _C{ 1,2,3 }, _C{ 1,2,3 } )
14.0
magnitude
abs(matrix)
Returns magnitude of one-dimensional matrix/vector
abs( _R{ 1,2,3 } )
abs( _C{ 1,2,3 } )
3.74
LUPQ decomp
LUPQ_decomp(matrix)
Returns LUPQ decomposition of a square matrix A as a tuple of 4 matrixes where PAQ = LU




LUP decomp
LUP_decomp(matrix)
Returns LUP decomposition of a square matrix as a tuple of 3 matrixes




QR decomp
qrdecomp(matrix)
Returns QR decomposition of a matrix as a tuple of 2 matrices




SVD decomp
SVD_decomp(matrix)
Returns SVD decomposition of a matrix as a tuple of 3 matrixes




eigenvalues/vectors
eigen(matrix)
Returns eigenvalues and eigenvalues of a square matrix as a vector of pairs




identity matrix
identity_matrix(unsigned int)
Returns an identity matrix of the specified dimension
identity_matrix(3)
{ {1.0,0.0,0.0}, {0.0,1.0,0.0}, {0.0,0.0,1.0} }
NULL matrix
null_matrix(unsigned int, unsigned int)
Returns a NULL matrix of the specified (row, col) dimensions
null_matrix(2, 3)
NULL
zero matrix
zero_matrix(unsigned int, unsigned int)
Returns a zero matrix of the specified (row, col) dimensions
zero_matrix(2, 3)
{ {0.0,0.0,0.0}, {0.0,0.0,0.0} }
matrix dimension
matrix_dim(matrix)
Returns the dimensions of the specified matrix as a tuple of (row, col) integers. When the input matrix is NULL, the function returns NULL. 
matrix_dim(_R{1, 2, 3})
TUPLE<INT,INT>(1, 3)
Matrix Operators
Operator
Syntax
Purpose
<
matrix < matrix
Less than comparison
>
matrix > matrix
Greater than comparison
<=
matrix <= matrix
Less than or equal to comparison
>=
matrix >= matrix
Greater than or equal to comparison
=
matrix = matrix
Equal to comparison
<>
matrix <> matrix
Not equal to comparison
[,]
matrix[rowIndex,colIndex]
Access matrix at the given row and column. 1-Indexed. Accessing an NULL matrix or using a NULL index returns NULL. Access out of bounds returns a NULL.
[:,:]
matrix[r1:c1,r2:c2]
Slice matrix at the specified row range and column range.
The index starts at 1.
Slicing a NULL matrix or using a NULL index returns NULL.
Start indices omitted or below 1 become 1. End indices omitted or above the number of rows/columns become the number of rows/columns. Example — matrix_10X10[:10,-1:20] is equivalent to matrix[1:10,1:10].
Ranges completely out of bounds throw an INVALID_ARGUMENT error.
Sequential slices like matrix[1:1,1:1][1:1,1:1] throw an error.
Slices cannot be combined with access. matrix[1,1:10] is not allowed, but matrix_10X10[1:1,1:10] is.
Trying to slice with non-constant indexes throws an error.
Operator Examples
Operator
SQL Predicate
Result
<
{ {1,2,3}, {4,5,6} } < { {4,5,6}, {1,2,3} }
true
﻿
{ {1,2,3}, {4,5,6} } < { {1,2,2}, {1,2,3} }
false
﻿
{ {1,2,3}, {4,5,6} } < { {1,2,3}, {4,5,6} }
false
>
{ {4,5,6}, {4,5,6} } > { {1,2,3}, {7,8,9} }
true
﻿
{ {1,2,3}, {4,5,6} } > { {4,5,6}, {1,2,3} }
false
﻿
{ {1,2,3}, {4,5,6} } > { {1,2,3}, {4,5,6} }
false
<=
{ {1,2,3}, {4,5,6} } <= { {4,5,6}, {1,2,3} }
true
﻿
{ {1,2,3}, {4,5,6} } <= { {1,2,2}, {1,2,3} }
false
﻿
{ {1,2,3}, {4,5,6} } <= { {1,2,3}, {4,5,6} }
true
>=
{ {4,5,6}, {4,5,6} } >= { {1,2,3}, {7,8,9} }
true
﻿
{ {1,2,3}, {4,5,6} } >= { {1,5,6}, {1,2,3} }
false
﻿
{ {1,2,3}, {4,5,6} } >= { {1,2,3}, {4,5,6} }
true
=
{ {1,2,3}, {4,5,6} } = { {1,2,3}, {4,5,6} }
true
﻿
{ {1,2,3}, {4,5,6} } = { {1,2,4}, {4,5,6} }
false
<>
{ {1,2,3}, {4,5,6} } <> { {1,2,3}, {4,5,6} }
false
﻿
{ {1,2,3}, {4,5,6} } <> { {1,2,4}, {4,5,6} }
true
matrix_access_operator
{ {1,2,3}, {4,5,6} }[1,2]
2.0
﻿
{ {1,2,3}, {4,5,6} }[NULL,2]
NULL
matrix_slice_operator
{ {1,2,3}, {4,5,6} }[1:,2:3]
{ {2.0,3.0}, {5.0,6.0} }
﻿
{ {1,2,3}, {4,5,6} }[1:NULL,2:3]
NULL
Related Links
﻿Array Functions and Operators﻿﻿
﻿Math Functions and Operators﻿﻿
﻿Tuple Functions and Operators﻿﻿
﻿Data Types﻿﻿
﻿Query Ocient﻿﻿

Updated 10 May 2024

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TABLE OF CONTENTS

Matrix Functions

Matrix Operators

Operator Examples

Function	Syntax	Purpose	Example	Result
Constructor	make_matrix_ixj(i, j, e_00, …, e_ij)	Creates an ixj matrix with elements e_00, …, e_ij	make_matrix_1x3(1,3,1,2,3)	{ {1.0,2.0,3.0} }
Constructor (2)	matrix_from_text(string)	Creates a matrix from the specified string	matrix_from_text('{ {1,2,3}, {1,2,3} }')	{ {1.0,2.0,3.0}, {1.0,2.0,3.0} }
+ (matrix addition)	matrix + matrix	Performs matrix addition	{ {1,2,3}, {4,5,6} } + { {4,5,6}, {1,2,3} }	{ {5.0,7.0,9.0}, {5.0,7.0,9.0} }
- (matrix subtraction)	matrix - matrix	Performs matrix subtraction	{ {1,2,3}, {4,5,6} } - { {4,5,6}, {1,2,3} }	{ {-3.0,-3.0,-3.0}, {3.0,3.0,3.0} }
* (scalar multiplication)	numeric * matrix	Performs scalar multiplication	2 * { {4,5,6}, {1,2,3} }	{ {2.0,4.0,6.0}, {8.0,10.0,12.0} }
* (matrix multiplication)	matrix * matrix	Performs matrix multiplication	{ {1,2,3}, {4,5,6} } * { {1,2}, {3,4}, {5,6} }	{ {22.0,28.0}, {49.0,64.0} }
/ (scalar division)	matrix / numeric	Performs scalar division	{ {1,2,3}, {4,5,6} } / 2	{ {0.5,1.0,1.5}, {2.0,2.5,3.0} }
transpose	transpose(matrix)	Returns transpose of the matrix	transpose( { {1,2,3}, {4,5,6} } )	{ {1.0,4.0}, {2.0,5.0}, {3.0,6.0} }
determinant	det(matrix)	Returns determinant of the matrix as a double	det( { {1,2}, {3,4} } )	-2.0
inverse	inverse(matrix)	Returns inverse of a square, invertible matrix	inverse( { {1,2}, {3,4} } )	{ {-2.0, 1.0}, {1.5, -0.5} }
trace	matrix_trace(matrix)	Returns trace of a square matrix as a double	matrix_trace( { {1,2,3}, {4,5,6}, {7,8,9} } )	15.0
vector sum	vector_sum(matrix)	Returns sum of elements in a 1D matrix/vector	vector_sum( _R{ 1,2,3 } ) vector_sum( _C{ 1,2,3 } )	6.0
vector min	vector_min(matrix)	Returns minimum of elements in a 1D matrix/vector	vector_min( _R{ 1,2,3 } ) vector_min( _C{ 1,2,3 } )	1.0
vector max	vector_max(matrix)	Returns maximum of elements in a 1D matrix/vector	vector_max( _R{ 1,2,3 } ) vector_max( _C{ 1,2,3 } )	3.0
vector argmin	vector_argmin(matrix)	Returns the argmin of a 1D matrix or vector. The index starts at 0.	vector_argmin(_R{1, -1, 5, 0})	1
vector argmax	vector_argmax(matrix)	Returns the argmax of a 1D matrix or vector. The index starts at 0.	vector_argmax(_C{1, -1, 5, 0})	2
softmax	softmax(matrix)	Returns the softmax of a 1D matrix or vector.	softmax(_R{1, 1})	{{0.5, 0.5}}
cross entropy loss	cross_entropy_loss(matrix, matrix)	Returns the cross entropy loss of two one-dimensional matrixes or vectors.	cross_entropy_loss( _R{ 0.2, 0.5, 0.8 }, _C{ 0,1,0 } ) cross_entropy_loss( _C{ 0.2, 0.5, 0.8}, _R{ 0,1,0 } ) cross_entropy_loss( _R{ 0.2, 0.5, 0.8 }, _R{ 0,1,0 } ) cross_entropy_loss( _C{ 0.2, 0.5, 0.8 }, _C{ 0,1,0} )	0.69
log loss	log_loss(matrix, matrix)	Returns the log loss of two one-dimensional matrixes or vectors.	log_loss( _R{ 0.2, 0.5, 0.8 }, _C{ 0,1,0 } ) log_loss( _C{ 0.2, 0.5, 0.8}, _R{ 0,1,0 } ) log_loss( _R{ 0.2, 0.5, 0.8 }, _R{ 0,1,0 } ) log_loss( _C{ 0.2, 0.5, 0.8 }, _C{ 0,1,0} )	2.53
logits loss	logits_loss(matrix, matrix)	Returns the logits loss of two one-dimensional matrixes or vectors.	logits_loss( _R{ 0.2, 0.5, 0.8 }, _C{ 0,1,0 } ) logits_loss( _C{ 0.2, 0.5, 0.8}, _R{ 0,1,0 } ) logits_loss( _R{ 0.2, 0.5, 0.8 }, _R{ 0,1,0 } ) logits_loss( _C{ 0.2, 0.5, 0.8 }, _C{ 0,1,0} )	2.44
hinge loss	hinge_loss(matrix, matrix)	Returns the hinge loss of two one-dimensional matrixes or vectors.	hinge_loss( _R{ 0.2, 0.5, 0.8 }, _C{ 0,1,0 } ) hinge_loss( _C{ 0.2, 0.5, 0.8}, _R{ 0,1,0 } ) hinge_loss( _R{ 0.2, 0.5, 0.8 }, _R{ 0,1,0 } ) hinge_loss( _C{ 0.2, 0.5, 0.8 }, _C{ 0,1,0} )	3.5
dot product	dot(matrix, matrix)	Returns dot product of two one-dimensional matrixes/vectors	dot( _R{ 1,2,3 }, _C{ 1,2,3 } ) dot( _C{ 1,2,3 }, _R{ 1,2,3 } ) dot( _R{ 1,2,3 }, _R{ 1,2,3 } ) dot( _C{ 1,2,3 }, _C{ 1,2,3 } )	14.0
magnitude	abs(matrix)	Returns magnitude of one-dimensional matrix/vector	abs( _R{ 1,2,3 } ) abs( _C{ 1,2,3 } )	3.74
LUPQ decomp	LUPQ_decomp(matrix)	Returns LUPQ decomposition of a square matrix A as a tuple of 4 matrixes where PAQ = LU
LUP decomp	LUP_decomp(matrix)	Returns LUP decomposition of a square matrix as a tuple of 3 matrixes
QR decomp	qrdecomp(matrix)	Returns QR decomposition of a matrix as a tuple of 2 matrices
SVD decomp	SVD_decomp(matrix)	Returns SVD decomposition of a matrix as a tuple of 3 matrixes
eigenvalues/vectors	eigen(matrix)	Returns eigenvalues and eigenvalues of a square matrix as a vector of pairs
identity matrix	identity_matrix(unsigned int)	Returns an identity matrix of the specified dimension	identity_matrix(3)	{ {1.0,0.0,0.0}, {0.0,1.0,0.0}, {0.0,0.0,1.0} }
NULL matrix	null_matrix(unsigned int, unsigned int)	Returns a NULL matrix of the specified (row, col) dimensions	null_matrix(2, 3)	NULL
zero matrix	zero_matrix(unsigned int, unsigned int)	Returns a zero matrix of the specified (row, col) dimensions	zero_matrix(2, 3)	{ {0.0,0.0,0.0}, {0.0,0.0,0.0} }
matrix dimension	matrix_dim(matrix)	Returns the dimensions of the specified matrix as a tuple of (row, col) integers. When the input matrix is NULL, the function returns NULL.	matrix_dim(_R{1, 2, 3})	TUPLE<INT,INT>(1, 3)

Operator	Syntax	Purpose
<	matrix < matrix	Less than comparison
>	matrix > matrix	Greater than comparison
<=	matrix <= matrix	Less than or equal to comparison
>=	matrix >= matrix	Greater than or equal to comparison
=	matrix = matrix	Equal to comparison
<>	matrix <> matrix	Not equal to comparison
[,]	matrix[rowIndex,colIndex]	Access matrix at the given row and column. 1-Indexed. Accessing an NULL matrix or using a NULL index returns NULL. Access out of bounds returns a NULL.
[:,:]	matrix[r1:c1,r2:c2]	Slice matrix at the specified row range and column range. The index starts at 1. Slicing a NULL matrix or using a NULL index returns NULL. Start indices omitted or below 1 become 1. End indices omitted or above the number of rows/columns become the number of rows/columns. Example — matrix_10X10[:10,-1:20] is equivalent to matrix[1:10,1:10]. Ranges completely out of bounds throw an INVALID_ARGUMENT error. Sequential slices like matrix[1:1,1:1][1:1,1:1] throw an error. Slices cannot be combined with access. matrix[1,1:10] is not allowed, but matrix_10X10[1:1,1:10] is. Trying to slice with non-constant indexes throws an error.

Operator	SQL Predicate	Result
<	{ {1,2,3}, {4,5,6} } < { {4,5,6}, {1,2,3} }	true
	{ {1,2,3}, {4,5,6} } < { {1,2,2}, {1,2,3} }	false
	{ {1,2,3}, {4,5,6} } < { {1,2,3}, {4,5,6} }	false
>	{ {4,5,6}, {4,5,6} } > { {1,2,3}, {7,8,9} }	true
	{ {1,2,3}, {4,5,6} } > { {4,5,6}, {1,2,3} }	false
	{ {1,2,3}, {4,5,6} } > { {1,2,3}, {4,5,6} }	false
<=	{ {1,2,3}, {4,5,6} } <= { {4,5,6}, {1,2,3} }	true
	{ {1,2,3}, {4,5,6} } <= { {1,2,2}, {1,2,3} }	false
	{ {1,2,3}, {4,5,6} } <= { {1,2,3}, {4,5,6} }	true
>=	{ {4,5,6}, {4,5,6} } >= { {1,2,3}, {7,8,9} }	true
	{ {1,2,3}, {4,5,6} } >= { {1,5,6}, {1,2,3} }	false
	{ {1,2,3}, {4,5,6} } >= { {1,2,3}, {4,5,6} }	true
=	{ {1,2,3}, {4,5,6} } = { {1,2,3}, {4,5,6} }	true
	{ {1,2,3}, {4,5,6} } = { {1,2,4}, {4,5,6} }	false
<>	{ {1,2,3}, {4,5,6} } <> { {1,2,3}, {4,5,6} }	false
	{ {1,2,3}, {4,5,6} } <> { {1,2,4}, {4,5,6} }	true
matrix_access_operator	{ {1,2,3}, {4,5,6} }[1,2]	2.0
	{ {1,2,3}, {4,5,6} }[NULL,2]	NULL
matrix_slice_operator	{ {1,2,3}, {4,5,6} }[1:,2:3]	{ {2.0,3.0}, {5.0,6.0} }
	{ {1,2,3}, {4,5,6} }[1:NULL,2:3]	NULL