What is the difference between orthogonal and orthonormal bases?

In an orthogonal basis, different basis vectors are perpendicular to each other, but they can have arbitrary lengths. In an orthonormal basis, the vectors are not only orthogonal but also have unit length. Gram–Schmidt first produces an orthogonal basis and then normalizes each vector to obtain an orthonormal basis.

Is Gram–Schmidt numerically stable?

The classical Gram–Schmidt algorithm can suffer from numerical instability in floating-point arithmetic, especially when vectors are nearly linearly dependent. A more stable variant is modified Gram–Schmidt, which reorders operations to reduce rounding errors. For small dimensions and moderate conditioning, classical Gram–Schmidt is usually adequate.

Gram–Schmidt Orthonormalization Calculator (Step‑by‑Step)

What is the Gram–Schmidt process?

The Gram–Schmidt process is an algorithm in linear algebra that takes a set of linearly independent vectors \(\{v_1,\dots,v_k\}\) in an inner product space (typically \(\mathbb{R}^n\)) and constructs an orthogonal (or orthonormal) set of vectors \(\{u_1,\dots,u_k\}\) (or \(\{e_1,\dots,e_k\}\)) that spans the same subspace.

This is fundamental for:

Building orthonormal bases of subspaces
QR factorization of matrices
Least squares and projections onto subspaces
Numerical methods and algorithms in scientific computing

Formulas of the Gram–Schmidt orthogonalization

Given linearly independent vectors \(\{v_1,\dots,v_k\}\) in \(\mathbb{R}^n\), define:

First vector: \[ u_1 = v_1 \]

For \(j \ge 2\): \[ u_j = v_j - \sum_{i=1}^{j-1} \operatorname{proj}_{u_i}(v_j) \] where the projection of \(v_j\) onto \(u_i\) is \[ \operatorname{proj}_{u_i}(v_j) = \frac{\langle v_j, u_i \rangle}{\langle u_i, u_i \rangle} u_i. \]

The vectors \(\{u_1,\dots,u_k\}\) are orthogonal and span the same subspace as the original vectors.

From orthogonal to orthonormal basis

To obtain an orthonormal basis \(\{e_1,\dots,e_k\}\), simply normalize each \(u_j\):

\[ e_j = \frac{u_j}{\|u_j\|} = \frac{u_j}{\sqrt{\langle u_j, u_j \rangle}}, \quad j=1,\dots,k. \]

Worked example

Consider the vectors in \(\mathbb{R}^3\):

\[ v_1 = \begin{bmatrix}1\\1\\0\end{bmatrix},\quad v_2 = \begin{bmatrix}1\\0\\1\end{bmatrix},\quad v_3 = \begin{bmatrix}0\\1\\1\end{bmatrix}. \]

Step 1: \(u_1\)

\[ u_1 = v_1 = \begin{bmatrix}1\\1\\0\end{bmatrix}, \quad \|u_1\| = \sqrt{1^2 + 1^2 + 0^2} = \sqrt{2}. \]

Step 2: \(u_2\)

Compute the projection of \(v_2\) onto \(u_1\): \[ \operatorname{proj}_{u_1}(v_2) = \frac{\langle v_2, u_1 \rangle}{\langle u_1, u_1 \rangle} u_1. \] Here \[ \langle v_2, u_1 \rangle = 1\cdot 1 + 0\cdot 1 + 1\cdot 0 = 1,\quad \langle u_1, u_1 \rangle = 2. \] So \[ \operatorname{proj}_{u_1}(v_2) = \frac{1}{2} \begin{bmatrix}1\\1\\0\end{bmatrix} = \begin{bmatrix}1/2\\1/2\\0\end{bmatrix}. \] Then \[ u_2 = v_2 - \operatorname{proj}_{u_1}(v_2) = \begin{bmatrix}1\\0\\1\end{bmatrix} - \begin{bmatrix}1/2\\1/2\\0\end{bmatrix} = \begin{bmatrix}1/2\\-1/2\\1\end{bmatrix}. \]

Step 3: \(u_3\)

Subtract projections onto both \(u_1\) and \(u_2\): \[ u_3 = v_3 - \operatorname{proj}_{u_1}(v_3) - \operatorname{proj}_{u_2}(v_3). \] You can reproduce these steps with the calculator by loading the example and inspecting the detailed output.

When does Gram–Schmidt fail?

The algorithm assumes that the input vectors are linearly independent. If they are not, then at some step the vector \(u_j\) becomes the zero vector:

\[ u_j = v_j - \sum_{i=1}^{j-1} \operatorname{proj}_{u_i}(v_j) = 0. \]

In that case, \(\|u_j\| = 0\) and you cannot normalize it to obtain \(e_j\). The calculator detects this situation and warns you that your vectors are linearly dependent or numerically very close to dependent.

Classical vs. modified Gram–Schmidt

In exact arithmetic, classical and modified Gram–Schmidt produce the same orthonormal basis. In floating‑point arithmetic, classical Gram–Schmidt can lose orthogonality when vectors are nearly dependent. Modified Gram–Schmidt reorders the operations to improve numerical stability.

This calculator implements the classical algorithm with careful rounding for educational clarity. For small dimensions (e.g. \(n \le 6\)) and moderate conditioning, the results are accurate and easy to follow step by step.

FAQ about the Gram–Schmidt process

Do I need my vectors to be in \(\mathbb{R}^n\)?

The process works in any inner product space. This tool is specialized to real coordinate vectors in \(\mathbb{R}^n\) with the standard dot product \(\langle x,y\rangle = \sum_i x_i y_i\).

Can I use complex vectors?

The general theory extends to complex inner product spaces using the Hermitian inner product \(\langle x,y\rangle = \sum_i x_i \overline{y_i}\). This calculator currently assumes real entries only.

How is this related to QR factorization?

If you arrange your input vectors as the columns of a matrix \(A\), applying Gram–Schmidt to those columns produces an orthonormal matrix \(Q\) and an upper triangular matrix \(R\) such that \(A = QR\). This is the classical QR factorization.

What if I input more vectors than the dimension?

In \(\mathbb{R}^n\), any set of more than \(n\) vectors is automatically linearly dependent. The calculator will detect dependence during the process and stop with an error message.

Gram–Schmidt Orthonormalization Calculator

Gram–Schmidt Calculator

Orthogonal basis \(\{u_1,\dots,u_k\}\)

Orthonormal basis \(\{e_1,\dots,e_k\}\)

Step‑by‑step derivation