# Diagonal Matrices
- First of all, diagonal matrices are zero everywhere except the main diagonal
- But they can also be 0’s on the main diagonal too
- So mistake 1
- A diagonal matrix is sometimes called a [scaling matrix](https://en.wikipedia.org/wiki/Scaling_matrix), since matrix multiplication with it results in changing scale (size). Its determinant is the product of its diagonal values.
- Look at this scaling transformation done by Apple’s ARKit
- All done one the main diagonal
- Next mistake, yes you are right dot product → scalar
- While Hadamard product → vector
- In other words, dot product is the sum of the entries of the Hadamard product
- Sorry about that, fixed, thanks for your careful examination
[ios - What are the first two columns in SCNMatrix4 - Stack Overflow](https://www.notion.so/ios-What-are-the-first-two-columns-in-SCNMatrix4-Stack-Overflow-a887b873d7334b84a594d6b1441bc4a2)
[What are the first two columns in SCNMatrix4](https://stackoverflow.com/questions/48264518/what-are-the-first-two-columns-in-scnmatrix4)
SCNMatrix4 is the 3d [transformation matrix](https://en.wikipedia.org/wiki/Transformation_matrix). In short:
```
M = T * R * S
```
Translation by (tx, ty, tz):
```
┌ ┐
T = | 1 0 0 tx |
| 0 1 0 ty |
| 0 0 1 tz |
| 0 0 0 1 |
└ ┘
```
Scale by (sx, sy, sz):
```
┌ ┐
S = | sx 0 0 0 |
| 0 sy 0 0 |
| 0 0 sz 0 |
| 0 0 0 1 |
└ ┘
```
Rotation by (rx, ry, rz):
```
R = ZYX
┌ ┐
X = | 1 0 0 0 |
| 0 cos(rx) -sin(rx) 0 |
| 0 sin(rx) cos(rx) 0 |
| 0 0 0 1 |
└ ┘
┌ ┐
Y = | cos(ry) 0 sin(ry) 0 |
| 0 1 0 0 |
| -sin(ry) 0 cos(ry) 0 |
| 0 0 0 1 |
└ ┘
┌ ┐
Z = | cos(rz) -sin(rz) 0 0 |
| sin(rz) cos(rz) 0 0 |
| 0 0 1 0 |
| 0 0 0 1 |
└ ┘
```
