# Difference between column- and row vector matrix multiplication (vector transform)

Let's say we have a 4-by-4 matrix $$\A\$$ which represents some transformation.

We can use this matrix to transform a vector in two ways.

• $$\Ax\$$ by assuming $$\x\$$ is a 4-by-1 column vector.
• $$\xA\$$ by assuming $$\x\$$ is a 1-by-4 row vector.

Up until yesterday, I have not seen this $$\xA\$$ variant and I thought it was unnecessary. Then I came across this DirectX code from Microsoft.

This function takes just the projection matrix, computes the inverse projection matrix and then does this $$\xA\$$ variant (not $$\Ax\$$) to transform (or unproject) a bunch of coordinates.

At first, I wasn't looking and did $$\Ax\$$ which came out nonsensical and now I'm trying to understand why.

I've noticed that $$\Ax = xA^T\$$ but when I look at the way the matrix multiplication is carried out. The components that make up the result are very different depending on whether I use $$\Ax\$$ or $$\xA\$$ (as it should be, matrix multiplication is not commutative).

The analytic solution for $$\Ax\$$ (3-by-3 matrix and row vector)

$$\begin{pmatrix} x \\ y \\ z \end{pmatrix} \cdot \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} = \begin{pmatrix} ax + dy + gz \\ bx + ey + hz \\ cx + fy + iz \end{pmatrix}$$

The analytic solution for $$\xA\$$ (3-by-3 matrix and column vector)

$$\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \cdot \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} ax+by+cz \\ dx+ey+fz \\ gx+hy+iz \end{pmatrix}$$

I use Mathematica to test these things. On a hunch I tried the analyitic solution for $$\xA\$$ with the elements of $$\A\$$ transposed manually and of course, I get:

$$\begin{bmatrix} a & d & g \\ b & e & h \\ c & f & i \end{bmatrix} \cdot \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} ax+by+cz \\ dx+ey+fz \\ gx+hy+iz \end{pmatrix}$$

Which is equivalent to $$\Ax\$$ except we now have a column vector not a row vector. I'm assuming all of this is purely conventional (and depends on whether I'm assuming row- or column-major matrix layout). It looks like I might have answered the question my self but if I'm wrong about anything here please correct me.

• As you said, it is purely conventional. Some people prefer rows, other columns, it also depends on which library you use, but it does not change anything Commented Nov 25, 2016 at 12:30
• I can only recommend the excellent "Game Engine Architecture" by Jason Gregory, it explains very well all you need to know, including 3D Math Commented Nov 25, 2016 at 12:37
• Yes, almost all linear algebra uses the row major convention since matrices are operators and operators are usually applied in a left-to-right fashion; this is because operators, just like functions, act on certain quantities. It feels natural to say "apply this operator to that object" not vice-versa "on this object apply that operator". Both sentences are correct in both natural and mathematical language. But in the mathematical language, one has to switch the major axis (column or row) for both operand and operator (i.e. vector and matrix) and then switch their order of appearance. Commented Nov 25, 2016 at 13:32
• Row-major vs column-major is the one of main differences between OpenGL and DirectX. It seems OpenGL uses column-major, while DirectX uses row-major. It's pure conventional, but it starts to matter in what order to perform matrix vs vector multiplication in shader. In GLSL shader if use OpenGL based math library you would use "Ax" convention, but if you use DirectX based math library for your OpenGL app, you would need to use "xA convention". Also this makes difference in what order you need to multiply your matrices. Commented Nov 25, 2016 at 16:00
• In OpenGL based math libraries it's mat4.mult(ProjectionMatrix, mat4.mult(ViewMatrix ,ModelMatrix)), while in DirectX based math libraries it's mat4.mult(mat4.mult(ModelMatrix,ViewMatrix),ProjectionMatrix) Commented Nov 25, 2016 at 16:02

Since I'm using a row-major convention for my matrix math library, transforming a vector must be implemented as xA. If I use a column-major convention I implement vector transform as Ax.
• it is worth noting that A: HLSL itself is in column-major and B: D3D (atleast D3D11) lets you compile shaders with D3DCOMPILE_PACK_MATRIX_COLUMN_MAJOR. Which omits the transposing on the CPU (with a minimal gpu cost) Commented Nov 11, 2020 at 14:16