Gram-Schmidt Orthogonalization Algorithm Explained

📚 Understanding the Gram-Schmidt Orthogonalization Algorithm

The Gram-Schmidt process is a method for orthogonalizing a set of vectors in an inner product space, most commonly Euclidean space $\mathbb{R}^n$. In simpler terms, it takes a set of linearly independent vectors and turns them into a set of orthogonal vectors that span the same subspace. Orthogonal vectors are vectors that are perpendicular to each other.

📜 History and Background

The algorithm is named after Jørgen Pedersen Gram and Erhard Schmidt, although it appeared earlier in the work of Laplace and Cauchy. Gram published his method in 1883, while Schmidt presented a more general version in 1907. It's a cornerstone in linear algebra and has applications in various fields like signal processing and numerical analysis.

🔑 Key Principles

• 📐 Projection: The core idea is to project one vector onto another and subtract that projection. This ensures the resulting vector is orthogonal to the vector it was projected onto.
• ➕ Iteration: The process is iterative. You orthogonalize the first vector, then use that result to orthogonalize the second, and so on.
• ✨ Normalization (Optional): After orthogonalizing, you can normalize the vectors (divide by their magnitudes) to create an orthonormal basis, where each vector has a length of 1.

🧮 The Algorithm

Given a set of linearly independent vectors {$v_1, v_2, ..., v_n$}, the Gram-Schmidt process constructs an orthogonal basis {$u_1, u_2, ..., u_n$} as follows:

1. $u_1 = v_1$
2. $u_2 = v_2 - \text{proj}_{u_1}(v_2) = v_2 - \frac{\langle v_2, u_1 \rangle}{\langle u_1, u_1 \rangle}u_1$
3. $u_3 = v_3 - \text{proj}_{u_1}(v_3) - \text{proj}_{u_2}(v_3) = v_3 - \frac{\langle v_3, u_1 \rangle}{\langle u_1, u_1 \rangle}u_1 - \frac{\langle v_3, u_2 \rangle}{\langle u_2, u_2 \rangle}u_2$
4. And so on, until: $u_n = v_n - \sum_{i=1}^{n-1} \frac{\langle v_n, u_i \rangle}{\langle u_i, u_i \rangle}u_i$

Where $\langle x, y \rangle$ denotes the inner product (dot product) of vectors $x$ and $y$, and $\text{proj}_{u}(v)$ is the projection of vector $v$ onto vector $u$.

➗ Normalization

To obtain an orthonormal basis {$e_1, e_2, ..., e_n$}, normalize each vector:

$e_i = \frac{u_i}{||u_i||}$, where $||u_i||$ is the magnitude (or length) of $u_i$.

💡 Real-world Example

Let's say we have two vectors in $\mathbb{R}^2$: $v_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$ and $v_2 = \begin{bmatrix} 2 \\ 1 \end{bmatrix}$.

1. $u_1 = v_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$
2. $u_2 = v_2 - \frac{\langle v_2, u_1 \rangle}{\langle u_1, u_1 \rangle}u_1 = \begin{bmatrix} 2 \\ 1 \end{bmatrix} - \frac{3}{2}\begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0.5 \\ -0.5 \end{bmatrix}$

So, $u_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$ and $u_2 = \begin{bmatrix} 0.5 \\ -0.5 \end{bmatrix}$ are orthogonal.

To orthonormalize, we divide by their magnitudes:

1. $e_1 = \frac{u_1}{||u_1||} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} \end{bmatrix}$
2. $e_2 = \frac{u_2}{||u_2||} = \frac{1}{\sqrt{0.5}}\begin{bmatrix} 0.5 \\ -0.5 \end{bmatrix} = \begin{bmatrix} \frac{\sqrt{2}}{2} \\ -\frac{\sqrt{2}}{2} \end{bmatrix}$

Thus, {$e_1, e_2$} form an orthonormal basis.

📝 Conclusion

The Gram-Schmidt process is a fundamental tool for creating orthogonal or orthonormal bases from any set of linearly independent vectors. It's widely used in various mathematical and computational applications. Understanding it provides a solid foundation for more advanced topics in linear algebra.