UnityGame/Library/PackageCache/com.unity.mathematics/Documentation~/4x4-matrices.md

4×4 matrices

To create a 4×4 transformation matrix, use the constructors in float4x4 to assign one value to all elements of the matrix, or individually set all 16 elements directly:

// Unity Mathematics example
void Build4x4UnityMathematics()
{
   var c0 = new float4(1.0f, 0.0f, 0.0f, 0.0f);
   var c1 = new float4(0.0f, 1.0f, 0.0f, 0.0f);
   var c2 = new float4(0.0f, 0.0f, 1.0f, 0.0f);
   var c3 = new float4(0.0f, 0.0f, 0.0f, 1.0f);
   var m = new float4x4(c0, c1, c2, c3);
}

Multiplying a 4×4 matrix

Unity Mathematics and UnityEngine define the * operator differently. The * operator for float4x4 implements componentwise multiplication. If you multiply a float4x4 of all 1s with 0.5 on the diagonal, you get back the half identity because the upper and lower triangles of the matrix are multiplied by the respective zero entries from f4x4_HalfIdentity:

// Unity Mathematics example
void OperatorMultiply4x4UnityMathematics()
{
   float4x4 result = f4x4_Ones * f4x4_HalfIdentity;
   // result:
   // 0.5, 0.0, 0.0, 0.0,
   // 0.0, 0.5, 0.0, 0.0,
   // 0.0, 0.0, 0.5, 0.0,
   // 0.0, 0.0, 0.0, 0.5
}

Multiplying a 4×4 matrix and a 4D vector

Use the math.mul method to multiply a 4×4 matrix and a 4D vector. If you supply a float4x4 as the first parameter and a float4 as the second, it performs a 4×4 matrix multiplication with a 4×1 column vector, which returns a 4×1 column vector as a float4.

math.mul can also multiply a 1×4 row vector by a 4×4 matrix to produce a 1×4 row vector by taking a float4 as the first parameter and a float4×4 as the second. Unity Mathematics stores the row vector in a float4 and it isn't treated as a separate type.

// Unity Mathematics example
void Multiply4x4AndVector4UnityMathematics()
{
   float4 result1 = math.mul(f4x4, f4); // 4x4 * 4x1 = 4x1
   float4 result2 = math.mul(f4, f4x4); // 1x4 * 4x4 = 1x4
}