Orthogonalize a set of machine-precision vectors:

Orthogonalize complex vectors:

Orthogonalize exact vectors:

Orthogonalize arbitrary-precision vectors:

Orthogonalize symbolic vectors:

Simplify the result assuming a and b are real-valued:

Large numerical matrices are handled efficiently:

Orthogonalize the rows of a sparse matrix:

Orthogonalize the rows of structured matrices:

Orthogonalizing a diagonal matrix produces another diagonal matrix:

Orthogonalize HilbertMatrix:

Find a symbolic basis, assuming all variables are real:

Orthogonalize vectors that are not lists using an explicit inner product:

Specify the inner product using a pure function:

Below the tolerance, two vectors are not recognized as linearly independent:

m forms a set of vectors that are nearly linearly dependent:

Deviation from orthonormality for the default method:

Deviation for all of the methods:

For a large numerical matrix, the Householder method is usually fastest:

Project the vector on the plane spanned by the vectors and :

Construct an orthonormal basis that spans the same space as the :

The projection in the plane is the sum of the projections onto :

Find the component perpendicular to the plane:

Confirm the result by projecting onto the normal to the plane:

Visualize the plane, the vector and its parallel and perpendicular components:

The Frenet–Serret system encodes every space curve's properties in a vector basis and scalar functions. Consider the following curve:

Construct an orthonormal basis from the first three derivatives using Orthogonalize:

Ensure that the basis is right-handed:

Compute the curvature, , and torsion, , which quantify how the curve bends:

Verify the answers using FrenetSerretSystem:

Visualize the curve and the associated moving basis, also called a frame:

Find the orthogonal projection of the vector onto the space spanned by the vectors , and :

First, construct an orthonormal basis for the space:

The component in the space is given by :

The difference is perpendicular to any vector in the span of the :

If the linear system has no solution, the best approximate solution is the least-squares solution. That is the solution to , where is the orthogonal projection of onto the column space of . Consider the following and :

The linear system is inconsistent:

Find an orthonormal basis for the space spanned by the columns of :

Compute the orthogonal projection of onto the spaced spanned by the :

Visualize , its projections onto the and :

Solve :

Confirm the result using LeastSquares:

Orthogonalize can be used to find a best-fit curve to data. Consider the following data:

Extract the and coordinates from the data:

Let have the columns and , so that minimizing will be fitting to a line :

Get the coefficients and for a linear least‐squares fit:

Verify the coefficients using Fit:

Plot the best-fit curve along with the data:

Find the best-fit parabola to the following data:

Extract the and coordinates from the data:

Let have the columns , and , so that minimizing will be fitting to :

Construct orthonormal vectors that have the same column space as :

Get the coefficients , and for a least‐squares fit:

Verify the coefficients using Fit:

Plot the best-fit curve along with the data:

Find an orthonormal basis for the column space of the following matrix , and then use that basis to find a QR factorization of :

Apply Gram–Schmidt to the columns of , then define as the matrix whose columns are those vectors:

Let :

Confirm that :

Compare with the result given by QRDecomposition; the matrices are the same:

The matrices differ by a transposition because QRDecomposition gives the row-orthonormal result:

For a Hermitian matrix (more generally, any normal matrix), the eigenvectors are orthogonal, and it is conventional to define the projection matrices , where is a normalized eigenvector. Show that the action of the projection matrices on a general vector is the same as projecting the vector onto the eigenspace for the following matrix :

Verify the is Hermitian:

Find the eigenvalues and eigenvectors:

and are both orthogonal to since they come from different eigenspaces:

They need not and are not orthogonal to each other, since they have the same eigenvalue:

Use Orthogonalize to create an orthonormal basis out of the :

Compute the projection matrices:

Confirm that multiplying a general vector by equals the projection of the vector onto :

Since the form an orthonormal basis, the sum of the must be the identity matrix:

Moreover, the sum of is the original matrix :

A positive-definite, real symmetric matrix or metric defines an inner product by :

Being positive-definite means that the associated quadratic form is positive for :

Note that Dot itself is the inner product associated with the identity matrix:

Orthogonalize the standard basis of to find an orthonormal basis:

Confirm that this basis is orthonormal with respect to the inner product :

Fourier series are projections onto an orthonormal basis in the inner product space . Define the standard inner product on square-integrable functions:

Orthogonalize functions of the form in this inner product:

equals the symmetric Fourier coefficient corresponding to FourierParameters{0,1}:

Confirm using FourierCoefficient:

The Fourier series is the projection of onto the space spanned by the :

Confirm the result using FourierSeries:

LegendreP defines a family of orthogonal polynomials with respect to the inner product . Orthogonalize the polynomials for from zero through four to compute scalar multiples of the first five Legendre polynomials:

Compare to the conventional Legendre polynomials:

For each , and differ by a factor of :

HermiteH defines a family of orthogonal polynomials with respect to the inner product . Apply the unnormalized Gram–Schmidt process to the polynomials for from zero through four to compute scalar multiples of the first five Hermite polynomials:

Compare to the conventional Hermite polynomials:

For each , and differ by a constant multiple: