Machine-precision eigenvalue decomposition:

Approximate 18-digit-precision eigenvalue decomposition:

Compute a decomposition of a complex matrix:

Check the result in machine arithmetic:

A real, exact eigendecomposition:

A complex, exact eigendecomposition:

Eigenvalue decomposition of a symbolic matrix:

The eigenvalue decomposition of large numerical matrices is computed efficiently:

Eigenvalue decomposition of sparse matrices:

Eigendecompositions of structured matrices:

The units of a QuantityArray object in the diagonal matrix, leaving the transformation matrix dimensionless:

IdentityMatrix[n] has a trivial eigendecomposition, up to the ordering of the eigenvectors:

Eigenvectors of HilbertMatrix:

If the matrix is first numericized, the matrix (but not the matrix) changes significantly:

This is because the eigenvectors are normalized for numeric input:

Eigenvalue decomposition of a CenteredInterval matrix:

Check that the relation gives small intervals containing zero for all components:

A 3×3 Vandermonde matrix:

In general, for exact 3×3 matrices, the result will be given in terms of Root objects:

To get the result in terms of radicals, use the Cubics option:

A matrix with a repeated eigenvalue that does not have a full set of eigenvectors:

At machine precision and automatic tolerance, this does not have an eigenvalue decomposition:

Add random noise to the matrix:

At default tolerance, this will give an eigenvalue decomposition:

Set a tolerance somewhat lower than the level of the added noise:

A 4×4 matrix:

In general, for a 4×4 matrix, the result will be given in terms of Root objects:

You can get the result in terms of radicals using the Cubics and Quartics options:

In {t,v}=EigenvalueDecomposition[m], the columns of t are eigenvectors of m:

This means the columns of m.t will equal the columns of t multiplied by the corresponding eigenvectors:

Let v₁ and v₂ be the eigenvectors

Eigenvectors with positive eigenvalues point in the same direction when acted on by the matrix:

Eigenvectors with negative eigenvalues point in the opposite direction when acted on by the matrix:

Consider the following matrix and its associated quadratic form :

The eigenvectors are the axes of the hyperbolas defined by :

The sign of the eigenvalue corresponds to the sign of the right-hand side of the hyperbola equation:

Here is a positive-definite quadratic form in three dimensions:

Plot the surface :

Get the symmetric matrix for the quadratic form, using CoefficientArrays:

Numerically compute its eigendecomposition:

Show the principal axes of the ellipsoid:

Consider now the quadratic form defined by d:

It has level sets of the same dimensions as q:

However, the level sets are rotated to align with the coordinate axes:

Diagonalize the following matrix as :

This is given directly by EigenvalueDecomposition[m]:

Confirm the identity :

Any function of the matrix can now be computed as . For example, MatrixPower:

Similarly, MatrixExp becomes trivial, requiring only exponentiating the diagonal elements of :

Let be the linear transformation whose standard matrix is given by the matrix . Find a basis for with the property that the representation of in that basis is diagonal:

Compute the eigenvalue decomposition of :

Let consist of the columns of :

converts from -coordinates to standard coordinates, and its inverse converts in the reverse direction:

Thus is given by , which is diagonal:

Note that this is simply the matrix :

A real-valued symmetric matrix is orthogonally diagonalizable as , with diagonal and real valued and orthogonal. Verify that the following matrix is symmetric and then diagonalize it:

Compute the eigenvalue decomposition, which directly gives :

To ensure an orthogonal transformation matrix, it is necessary to normalize the columns in :

This matrix is orthogonal:

Verify that :

A matrix is called normal if . Normal matrices are the most general kind of matrix that can be diagonalized by a unitary transformation. All real symmetric matrices are normal because both sides of the equality are simply :

Show that the following matrix is normal, then diagonalize it:

Confirm using NormalMatrixQ:

Find the eigendecomposition:

Unlike a real symmetric matrix, the diagonal matrix for this matrix is complex valued:

Normalizing each column in gives a unitary matrix:

Confirm the diagonalization :

Produce the general solution of the dynamical system when is the following stochastic matrix:

Find the eigendecomposition:

The general solution is then , where is an arbitrary starting point:

Note that since is diagonal, elementwise powers and MatrixPower are the same operation:

Verify that satisfies the dynamical equation up to numerical rounding:

Solve the system of ODEs , , . First, construct the coefficient matrix for the right-hand side:

Find the eigendecomposition:

Let be the matrix where the diagonal elements of are multiplied by and exponentiated:

The general solution is , for three arbitrary starting values:

Verify the solution using DSolveValue:

Suppose a particle is moving in a planar force field and its position vector satisfies and , where and are as follows. Solve this initial problem for :

First, compute the eigenvalues and corresponding eigenvectors of :

The general solution is . Use LinearSolve to determine the coefficient vector :

Construct the appropriate linear combination of the eigenvectors:

Note that can be computed by exponentiating the diagonal elements of :

Verify the solution using DSolveValue:

The Lorenz equations:

Find the Jacobian for the right-hand side of the equations:

Find the equilibrium points:

Find the eigendecomposition of the Jacobian at the equilibrium point in the first octant:

A function that integrates backward from a small perturbation of pt in the direction dir:

Show the stable curve for the equilibrium point on the right:

Find the stable curve for the equilibrium point on the left:

Show the stable curves along with a solution of the Lorenz equations:

In quantum mechanics, states are represented by complex unit vectors and physical quantities by Hermitian linear operators. The eigenvalues represent possible observations and the squared modulus of the components with respect to eigenvectors the probabilities of those observations. For the spin operator and state given, find the possible observations and their probabilities:

Computing the eigendecomposition, the possible observations are :

Normalize the eigenvectors in order to compute a proper rotation matrix:

This matrix is unitary:

The adjoint of projects onto the eigenvectors, giving relative probabilities of for and for :

In quantum mechanics, the energy operator is called the Hamiltonian , and a state evolves according to the Schrödinger equation . Given the Hamiltonian for a spin-1 particle in constant magnetic field in the direction, find the state at time of a particle that is initially in the state representing :

Computing the eigensystem, the energy levels are and :

Normalize the eigenvectors:

This matrix is unitary:

As , the state at time is given by:

The moment of inertia is a real symmetric matrix that describes the resistance of a rigid body to rotating in different directions. The eigenvalues of this matrix are called the principal moments of inertia, and the corresponding eigenvectors (which are necessarily orthogonal) the principal axes. Find the principal moments of inertia and principal axes for the following tetrahedron:

First compute the moment of inertia:

Compute the principal moments and axes:

Verify that the axes are orthogonal:

The center of mass of the tetrahedron is at the origin:

Visualize the tetrahedron and its principal axes: