THE IDEA
An oblong of terms
James Joseph Sylvester first used the word matrix in 1850, for an oblong arrangement of terms. Rows run across, columns run down.
See one at workA matrix is numbers in rows and columns.
It looks like a small spreadsheet, but it comes with rules for adding and multiplying. Those rules let one grid stand for a whole action, such as turning, stretching or moving a shape on screen.
- Rows and columns
- Rules for multiplying
- One grid, one move
- The idea behind 3D graphics
THE SHORT ANSWER
Numbers in a grid,
with rules attached.
A matrix is a way of keeping related numbers together.
Its rules turn the grid into a tool.
GRAPHICS
Every move in 16 numbers
Graphics courses teach that all of a coordinate transformation fits in a 4×4 matrix: rotation, scaling and moving a shape, all in one grid.
How it works
HISTORY
Older than the word
The Nine Chapters on the Mathematical Art, from China between 200 BC and 100 BC, set equation numbers out as a table on a counting board.
See the dates
FOUR STEPS TO READ ONE
Read a matrix
like a map.
You do not need any software. A pencil and squared paper are enough to see how a matrix changes a point.
Count the rows, then the columns
Size is always given rows first. A 4×4 graphics matrix has four of each, so 16 numbers.
Write the point as a column
Stack x, y and z in a column and add a 1 underneath. That 1 marks it as a point rather than a direction.
Multiply row by column
Each new coordinate comes from one row of the matrix: multiply it number by number with the column, then add up.
Mind the order
When several moves are chained, the multiplications are applied right to left, so swapping the order can change the result.
TWO NUMBERS WORTH KNOWING
An old method,
a modern grid.
One figure from the history of matrix methods,
one from how screens draw 3D shapes today.
200–100BC
When the Nine Chapters was written
St Andrews' MacTutor history archive dates the Nine Chapters to the Han Dynasty, between 200 BC and 100 BC, and calls its counting-board table the first known example of matrix methods.
HISTORY · MACTUTOR
16 numbers
Numbers in a graphics transformation
A 4×4 matrix has 16 values. The x, y and z axes fill the first three columns and the position of the origin fills the last one.
GRAPHICS · NYU
The extra fourth coordinate is called a homogeneous coordinate. Points have it set to 1 and directions to 0, which is why sliding a shape moves its points but not its directions.
A QUICK TRANSFORMATION CHECK
Which part of the grid
does the work?
Pick a move to see which part of a 4×4 graphics matrix it uses. The bar shows how much of the grid the move changes.
Based on university computer graphics course notes from NYU and UC Berkeley.
A SHORT HISTORY
The word came
after the method.
Chinese mathematicians used a table of coefficients on a counting board over 2,000 years ago. Seki in Japan (1683) and Leibniz in Europe (1693) worked with determinants, and Gauss used elimination in 1803 to 1809 to study the orbit of the asteroid Pallas.
Sylvester named the matrix in 1850. In 1858 Arthur Cayley's Memoir on the Theory of Matrices gave the first abstract definition and an algebra for adding, multiplying and inverting them, described by the mathematician Nicholas Higham as the first paper on matrix algebra.
In conversation
Questions people ask
What is a matrix, simply?
A rectangular grid of numbers arranged in rows and columns. Sylvester, who coined the word in 1850, called it an oblong arrangement of terms. What makes it useful are the rules for combining matrices.
Why do computer graphics use 4×4 matrices?
Course notes from NYU say all the information of a 3D coordinate transformation can be placed in a 4×4 matrix. The extra row and column let a slide (translation) be written as a matrix alongside rotation and scaling.
Does the order of multiplying matrices matter?
Yes. UC Berkeley's graphics notes say a sequence of transformation matrices is applied right to left, so the same moves in a different order can give a different result.
Who invented matrices?
No single person. The Nine Chapters, from China between 200 BC and 100 BC, used a table of coefficients. Sylvester named the matrix in 1850, and Cayley's 1858 memoir set out matrix algebra.
Is a matrix the same as a spreadsheet?
They look alike, as both are grids of rows and columns. A matrix adds mathematical rules for adding, multiplying and inverting whole grids, which is what lets one matrix stand for an action.
Sources
Each figure on this page comes from the sources below.
- MacTutor History of Mathematics (University of St Andrews): Matrices and determinants
- Nicholas J. Higham: Cayley, Sylvester, and Early Matrix Theory (University of Manchester, 2007)
- NYU computer graphics class notes: Introduction to matrices (2015)
- UC Berkeley CS184 lecture 6 summary: 3D transformations