Add some documentation, Aff

README.md

@@ -1,15 +1,75 @@
 # Performant Embedded Algebra Kit (PEAK)
 
-
-> ⚠️ **Warning:** WORK IN PROGRESS and UNTESTED
-
-
 ## Description
 
-The Performant Embeddded Algebra Kit (PEAK) is a lightweight and easy-to-use library for performing various algebraic operations.
+The Performant Embedded Algebra Kit (PEAK) is a lightweight and easy-to-use library for performing various algebraic operations.
 
 ## Features
 
 - Matrix operations: Perform matrix addition, subtraction, multiplication, and inversion.
-- Kalman Filter implementation
+- Kalman Filter implementations
+    - basic Kalman Filter
+    - Dynamic Kalman Filter, with variable delta t
 
+## Usage
+
+### Basics
+
+The following example demonstrates the creation of matrices and basic operations on them.
+
+```cpp
+#include <sta/math/linalg/matrix.hpp>
+#include <sta/math/linalg/linalg.hpp>
+
+// Create a 3x2 matrix with every entry initialized to 0:
+sta::math::matrix m_0 = sta::math::matrix::zeros(3,2);
+
+// Set specific entries in the matrix:
+m_0.set(0, 1, 4); // set the entry in the first column and second column to 0
+m_0.set(1, 0, 5);
+m_0.set(2, 2, 99);
+
+// Display the matrix:
+m_0.show_serial();
+
+// Output:
+// Matrix shape: (3 x 2)
+// | 0.000000 | 4.000000 |
+// | 5.000000 | 0.000000 |
+// | 0.000000 | 0.000000 |
+
+// Create a 3x2 matrix with every entry initialized to 10:
+sta::math::matrix m_10 = sta::math::matrix::full(3,2,10); 
+
+// Matrices of the same shape can be added together using the + operator:
+sta::math::matrix m_a = m_0 + m_10;
+
+// m_a:
+// | 10.000000 | 14.000000 |
+// | 15.000000 | 10.000000 |
+// | 10.000000 | 10.000000 |
+
+// Create an identity matrix of size 3 and multiply it by 5:
+sta::math::matrix m_i = sta::math::matrix::eye(3) * 5;
+
+// Set a specific entry in the identity matrix:
+m_i.set(0, 2, 0.5);
+
+// m_i:
+// | 5.000000 | 0.000000 | 0.500000 |
+// | 0.000000 | 5.000000 | 0.000000 |
+// | 0.000000 | 0.000000 | 5.000000 |
+
+// Matrices of the appropriate shape can be multiplied together using the * operator:
+sta::math::matrix m_res = m_i * m_a;
+
+// m_res:
+// | 55.000000 | 75.000000 |
+// | 75.000000 | 50.000000 |
+// | 50.000000 | 50.000000 |
+
+```
+
+## Limitations
+
+As of now, only matrices with no more than 256 elements are supported.