assets altered

Author: prgyn8

.assets/matrix rec.mp4

@@ -1,43 +1,93 @@
+#### **A comprehensive implementation of core **Matrix Decomposition techniques** in Linear Algebra, providing an efficient computation approach, helps in uncovering hidden relationships in data, and lets you power many scalable applications in various fields of science and engineering.**
 
+______________________________________________________________________
 
-A comprehensive implementation of **_Matrix Decomposition Techniques_**, forming the foundations of various scientific and engineering applications. 
+To gain a deeper understanding of how Orthogonalization & Matrices Decomposition works in real-life applications, & how they save bunch of time through an approach of vectorization, you'll find such techniques used in;
 
-![alt text](.assets/matrix_giphy.gif)
+- **📡 Signal Processing**
+- **🤖 Control Systems and Robotics**
+- **🖼️ Image Processing**
+- **➗ Solving Linear Systems i.e. *AX = B***
 
-To gain a deeper understanding of how Orthogonalization & matrices Decomposition works in real-life applications, like,
-- **Signal Processing**
-- **Control Systems and Robotics**
-- **Solving Linear Systems i.e. *AX = B***
+With certain mathematical intuitions (*having visual introspections*),this project simplifies most of the abstract concepts and becomes easier to grasp and connect with practical applications.
 
-With mathematical intuition & visual introspection, the project makes most of the abstract concepts easier to grasp and connect with practical applications.
+![a snippet of notebook](.assets/matrix_snippet.gif)
 
-> This is the first draft of the project. More implementations—such as **Householder Reflection**, **Bidiagonalization**, **LU Decomposition**, and others—will be added soon.
+> You'll yet to see more implementations—such as **Householder Reflection**, **Bidiagonalization**, **LU Decomposition**, on this repo, and others—*these will be added soon*.
 
-## Features / What's Inside
+## What's Inside
 
-For a vector space, building *Orthonormal Basis* is the most essential part of Linear Algebra. 
+The **Gram-Schmidt Orthogonalization** is one of the fundamental process in Linear Algebra to achieve *Orthonormal Vectors* for a given vector space. The Orthonormal Basis are produced by iteratively removing vector projections — also known as the *Vector Projection Elimination method*.
 
-You can find top discussions about certain topics like Orthogonality, and why do we need it, ....here.
+**Terms like Orthogonality, QR Decomposition are being discussed in the — [🗨️Discussion section](https://github.com/PragyanTiwari/Matrix-Decompositions-Implementation-for-SVD-PCA/discussions).**
 
-#### Gram-Schmidt Orthogonalization
+Here's a snippet;
 
-One of the technique for creating Orthonormal Basis from a vector space (*assuming linear independence*) by using a Vector Projection Elimination method. 
+```bash
+def gs_Orthogonalization(X:np.ndarray)->np.ndarray:
+
+    Q = np.copy(X).astype("float64")
+    n_vecs = Q.shape[1]
+
+    # defining a function to compute the L2-norm
+    length = lambda x: np.linalg.norm(x)
+
+    # iteration with each vector in the matrix X
+    for nth_vec in range(n_vecs):
+
+        # iteratively removing each preceding projection from nth vector
+        for k_proj in range(nth_vec):
 
-Here's a snippet,
+            # the dot product would be the scaler coefficient 
+            scaler = Q[:,nth_vec] @ Q[:,k_proj]
+            projection = scaler * Q[:,k_proj]
+            Q[:,nth_vec] -= projection                 # removing the Kth projection
 
+        norm = length(Q[:,nth_vec])
+
+        # handling the case if the loop encounters linearly dependent vectors. 
+        # Since, they come already under the span of vector space, hence their value will be 0.
+        if np.isclose(norm,0, rtol=1e-15, atol=1e-14, equal_nan=False):
+            Q[:,nth_vec] = 0
+        else:
+            # making orthogonal vectors -> orthonormal
+            Q[:,nth_vec] = Q[:,nth_vec] / norm
+
+    return Q
+```
+
+To edit the notebook in a sandbox environment, run this;
+
+```bash
+uvx marimo edit --sandbox notebooks\Gram_Schmidt_QR_Decomposition.py
+```
 
 ## 🧪 Testing
 
-To test the export process, run `.github/scripts/build.py` from the root directory.
+The updates made on this project, can be tested for deployment, (and for personal experimentation) by the following;
+
+- Fork the repository.
+
+- Run uv sync to install dependencies (*uv lockfile will help*)
 
 ```bash
-uv run .github/scripts/build.py
+uv sync
 ```
 
-This will export all notebooks in a folder called `_site/` in the root directory. Then to serve the site, run:
+- To test the export process, we'll run `.github/scripts/build.py` from the root directory through a symlink.
 
 ```bash
-python -m http.server -d _site
+uv run build.py
 ```
 
-This will serve the site at `http://localhost:8000`.
+This will export all notebooks in a folder called `_site/` in the root directory
+
+## 🌱 Contribution Guide
+
+- If you find a bug or have a feature request, please open an [Issue](https://github.com/PragyanTiwari/Matrix-Decompositions-Implementation-for-SVD-PCA/issues).
+
+- PR will be reviewed by the maintainers.
+
+- Questions & Suggestions can be queried on the [Discussion section](https://github.com/PragyanTiwari/Matrix-Decompositions-Implementation-for-SVD-PCA/discussions).
+
+______________________________________________________________________

.assets/matrix_snippet.gif

@@ -2,8 +2,8 @@
 # requires-python = ">=3.12"
 # dependencies = [
 #     "marimo",
-#     "matplotlib",
-#     "numpy",
+#     "matplotlib==3.10.5",
+#     "numpy==2.3.2",
 # ]
 # ///
 import marimo