Gram Schmidt Calculator
Our free linear algebra calculator solves gram schmidt problems. Get worked examples, visual aids, and downloadable results.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Gram Schmidt Calculator
Calculator
Adjust values & calculateEnter your values below. Every result is computed in your browser — no data is sent to any server.
Formula: u_k = v_k - sum_{j=1}^{k-1} proj(v_k, u_j), where proj(v, u) = (v dot u)/(u dot u) * u
Worked example — Orthonormal basis: e1 = [0.707, 0.707, 0], e2 = [0.408, -0.408, 0.816], e3 = [-0.577, 0.577, 0.577]
Formula
u_k = v_k - sum_{j=1}^{k-1} proj(v_k, u_j), where proj(v, u) = (v dot u)/(u dot u) * uEach vector u_k is obtained by subtracting from v_k its projections onto all previously computed orthogonal vectors u_1, ..., u_{k-1}. The result is perpendicular to all previous vectors. Normalizing gives orthonormal vectors e_k = u_k / ||u_k||.
Worked Examples
Example 1: Orthogonalizing Three 3D Vectors
Problem:Apply Gram-Schmidt to v1 = [1, 1, 0], v2 = [1, 0, 1], v3 = [0, 1, 1].
Solution:u1 = v1 = [1, 1, 0], ||u1|| = sqrt(2) e1 = [1/sqrt(2), 1/sqrt(2), 0] proj(v2, u1) = (1+0+0)/(1+1+0) = 1/2 u2 = [1, 0, 1] - (1/2)[1, 1, 0] = [1/2, -1/2, 1], ||u2|| = sqrt(3/2) e2 = [1/sqrt(6), -1/sqrt(6), 2/sqrt(6)] proj(v3, u1) = (0+1+0)/2 = 1/2, proj(v3, u2) = (0-1/2+1)/(3/2) = 1/3 u3 = [0,1,1] - (1/2)[1,1,0] - (1/3)[1/2,-1/2,1] = [-2/3, 2/3, 2/3] e3 = [-1/sqrt(3), 1/sqrt(3), 1/sqrt(3)]
Result:Orthonormal basis: e1 = [0.707, 0.707, 0], e2 = [0.408, -0.408, 0.816], e3 = [-0.577, 0.577, 0.577]
Example 2: Detecting Linear Dependence
Problem:Apply Gram-Schmidt to v1 = [1, 2, 3], v2 = [2, 4, 6], v3 = [1, 0, 1].
Solution:u1 = v1 = [1, 2, 3], ||u1|| = sqrt(14) proj(v2, u1) = (2+8+18)/14 = 28/14 = 2 u2 = [2, 4, 6] - 2*[1, 2, 3] = [0, 0, 0] v2 is linearly dependent on v1 (it is exactly 2*v1). u3 = v3 - proj(v3, u1) = [1,0,1] - (1+0+3)/14 * [1,2,3] = [1-2/7, -4/7, 1-6/7]
Result:Only 2 independent vectors found (rank = 2). v2 = 2*v1 is dependent.
Frequently Asked Questions
What is the Gram-Schmidt process?
The Gram-Schmidt process is an algorithm that takes a set of linearly independent vectors and produces a set of orthogonal (or orthonormal) vectors that span the same subspace. It works by iteratively subtracting the projections of each new vector onto all previously computed orthogonal vectors. The result is a set of mutually perpendicular vectors that are much easier to work with in computations. The process is named after Jorgen Pedersen Gram and Erhard Schmidt, though the concept predates them. It is a cornerstone of numerical linear algebra and is used in QR decomposition, least squares problems, and signal processing.
How does the Gram-Schmidt process work step by step?
The Gram-Schmidt process starts with the first vector unchanged: u1 = v1. For the second vector, subtract its projection onto u1: u2 = v2 - proj(v2, u1), where proj(v2, u1) = (v2 dot u1)/(u1 dot u1) * u1. For the third vector, subtract projections onto both u1 and u2: u3 = v3 - proj(v3, u1) - proj(v3, u2). This continues for each additional vector. After computing each orthogonal vector, normalize it by dividing by its length to get an orthonormal vector. The projection subtraction removes the component parallel to each previous vector, leaving only the perpendicular component.
What is the difference between orthogonal and orthonormal vectors?
Orthogonal vectors are mutually perpendicular, meaning the dot product of any two different vectors in the set is zero. Orthonormal vectors are orthogonal vectors that additionally have unit length (magnitude 1). To go from orthogonal to orthonormal, simply divide each vector by its magnitude. Orthonormal vectors have special computational advantages: the inverse of an orthonormal matrix equals its transpose, projections are simplified (no denominator needed), and coordinate representations are trivial. The Gram-Schmidt process can produce either orthogonal vectors or, with an additional normalization step, orthonormal vectors.
What is the QR decomposition and how does Gram-Schmidt relate to it?
The QR decomposition factors a matrix A into Q (an orthogonal or unitary matrix) and R (an upper triangular matrix) such that A = QR. The Gram-Schmidt process directly computes the QR decomposition: the orthonormal vectors become the columns of Q, and R is formed from the projection coefficients. Specifically, R_ij = (v_j dot e_i) for i <= j and R_ij = 0 for i > j. QR decomposition is crucial for solving least squares problems, computing eigenvalues (QR algorithm), and providing numerically stable solutions to linear systems. It is one of the most important matrix factorizations in numerical computing.
What happens if the input vectors are linearly dependent?
If the input vectors are linearly dependent, the Gram-Schmidt process detects this automatically. At some step, after subtracting all projections, the resulting vector will be the zero vector (or numerically very close to zero). This happens because the current input vector is already expressible as a linear combination of the previous vectors, so subtracting projections removes everything. When this occurs, the dependent vector is skipped, and the process continues with the remaining vectors. The number of nonzero orthogonal vectors produced equals the rank of the original set of vectors. This property makes Gram-Schmidt useful for determining the rank and finding an orthogonal basis for the column space.
What is the modified Gram-Schmidt process and why is it preferred?
The modified Gram-Schmidt (MGS) process rearranges the order of computations to improve numerical stability. In classical Gram-Schmidt, all projections for a vector are computed using the original orthogonal vectors. In MGS, after computing each projection and subtracting it, the vector is immediately updated before computing the next projection. This seemingly minor change significantly reduces the accumulation of rounding errors. Classical Gram-Schmidt can produce vectors that are far from orthogonal in floating-point arithmetic, while MGS maintains much better orthogonality. For practical numerical computation, MGS or Householder reflections are strongly preferred over classical Gram-Schmidt.
What is a projection in the context of Gram-Schmidt?
A projection of vector v onto vector u, written proj(v, u), is the component of v that lies along the direction of u. The formula is proj(v, u) = (v dot u)/(u dot u) * u. Geometrically, it is the shadow of v cast onto the line defined by u. The scalar coefficient (v dot u)/(u dot u) tells you how many copies of u make up this shadow. In Gram-Schmidt, subtracting the projection removes the component of the new vector that is parallel to a previously computed orthogonal vector, leaving only the perpendicular component. This is why the resulting vectors are guaranteed to be orthogonal to each other.
What are the applications of the Gram-Schmidt process?
The Gram-Schmidt process has wide applications across mathematics and engineering. In signal processing, it creates orthogonal filter banks for efficient signal decomposition. In statistics, it orthogonalizes predictor variables in regression to diagnose multicollinearity. In quantum mechanics, it constructs orthonormal basis states from general state vectors. In computer graphics, it re-orthogonalizes rotation matrices that drift due to numerical errors. In communications engineering, it designs orthogonal codes for CDMA systems. In numerical methods, it is the foundation of the Arnoldi and Lanczos algorithms for computing eigenvalues of large sparse matrices.
How do you verify that the output vectors are truly orthogonal?
To verify orthogonality, compute the dot product of every pair of output vectors. Each dot product should be zero (or very close to zero in floating-point arithmetic). For n orthogonal vectors, there are n(n-1)/2 pairs to check. Additionally, if orthonormal vectors are desired, verify that each vector has unit length (its dot product with itself equals 1). A convenient way to check all conditions at once is to compute Q-transpose times Q, which should equal the identity matrix for orthonormal vectors. Any deviation from the identity indicates loss of orthogonality, which may occur due to numerical errors in ill-conditioned problems.
Can Gram-Schmidt be applied to function spaces?
Yes, the Gram-Schmidt process generalizes beautifully to any inner product space, including function spaces. Instead of the standard dot product, you use the inner product for functions: the integral of f(x)*g(x)*w(x) over the domain, where w(x) is a weight function. Classical orthogonal polynomials like Legendre, Chebyshev, Hermite, and Laguerre polynomials can be derived by applying Gram-Schmidt to the monomials {1, x, x^2, ...} with appropriate weight functions and domains. In Fourier analysis, the orthogonal sine and cosine functions form an orthonormal basis for function spaces. This generalization is fundamental to approximation theory and spectral methods.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
Related Calculators
🧮Annulus Area Calculator
Calculate annulus area with inputs, formulas, and instant results.
🧮Area Calculator
Calculate area with inputs, formulas, and instant results.
🧮Area of a Rectangle Calculator
Calculate the area, perimeter, and diagonal of a rectangle. Find missing sides from known area. Convert between metric and imperial area units.
🧮Area of Crescent Calculator
Calculate area of crescent with inputs, formulas, and instant results.
🧮Center of Mass Calculator
Calculate center of mass with inputs, formulas, and instant results.
🧮Centroid Calculator
Calculate centroid with inputs, formulas, and instant results.
🧮Chord Length Calculator
Calculate chord length with inputs, formulas, and instant results.
🧮Conic Sections Calculator
Calculate conic sections with inputs, formulas, and instant results.