Find the eigenvalues of a machine-precision matrix:

Approximate 20-digit precision eigenvalues:

Eigenvalues of a complex matrix:

Exact eigenvalues:

The eigenvalues of large numerical matrices are computed efficiently:

The largest five eigenvalues:

Three smallest eigenvalues:

Find the four largest eigenvalues, or as many as there are if fewer:

Repeated eigenvalues are listed multiple times:

Repeats are considered when extracting a subset of the eigenvalues:

Generalized machine-precision eigenvalues:

Generalized exact eigenvalues:

Compute the result at finite precision:

Find the generalized eigenvalues of symbolic matrices:

Find the two smallest generalized eigenvalues:

Eigenvalues of sparse matrices:

Eigenvalues of structured matrices:

IdentityMatrix always has all-one eigenvalues:

Eigenvalues of HilbertMatrix:

Eigenvalues uses Root to compute exact eigenvalues:

Explicitly use the cubic formula to get the result in terms of radicals:

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:

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 eigenvalues and eigenvectors:

Show the principal axes of the ellipsoid:

Diagonalize the following matrix as . First, compute 's eigenvalues:

Construct a diagonal matrix from the eigenvalues:

Next, compute 's eigenvectors and place them in the columns of a matrix:

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:

Find the eigenvalues of :

Let consist of the eigenvectors, and let be the matrix whose columns are the elements of :

converts from -coordinates to standard coordinates. Its inverse converts in the reverse direction:

Thus is given by , which is diagonal:

Note that this is simply the diagonal matrix whose entries are the eigenvalues:

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:

Computing the eigenvalues, they are real, as expected:

The matrix has the eigenvalues on the diagonal:

Next, compute the eigenvectors of :

For an orthogonal matrix, it is necessary to normalize the eigenvectors before placing them in columns:

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:

The eigenvalues of a real normal matrix that is not also symmetric are complex valued:

Construct a diagonal matrix from the eigenvalues:

Compute the eigenvectors:

Normalizing the eigenvectors and putting them in columns gives a unitary matrix:

Confirm the diagonalization :

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

Find the eigenvalues and eigenvectors:

Construct a diagonal matrix whose entries are the exponential of :

Construct the matrix whose columns are the corresponding eigenvectors:

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 value problem for :

First, compute the eigenvalues and corresponding eigenvectors of :

The general solution of the system is . Use LinearSolve to determine the coefficients:

Construct the appropriate linear combination of the eigenvectors:

Verify the solution using DSolveValue:

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

Find the eigenvalues and eigenvectors, using Chop to discard small numerical errors:

The general solution is an arbitrary linear combination of terms of the form :

Verify that satisfies the dynamical equation up to numerical rounding:

The Lorenz equations:

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

Find the equilibrium points:

Find the eigenvalues and eigenvectors of the Jacobian at the one 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 eigenvalues, the possible observations are :

Find the eigenvectors and normalize them in order to compute proper projections:

The relative probabilities are for and for :

In quantum mechanics, the energy operator is called the Hamiltonian , and a state with energy 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 eigenvalues, the energy levels are and :

Find and normalize the eigenvectors:

The state at time is the sum of each eigenstate evolving according to the Schrödinger equation:

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 axis for the following tetrahedron:

First compute the moment of inertia:

The principle moments are the eigenvalues of :

The principle axes are the eigenvectors of :

Verify that the axes are orthogonal:

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

Visualize the tetrahedron and its principal axes:

A generalized eigensystem can be used to find normal modes of coupled oscillations that decouple the terms. Consider the system shown in the diagram:

By Hooke's law it obeys , . Substituting in the generic solution gives rise to the matrix equation , with the stiffness matrix and mass matrix as follows:

Find the eigenfrequencies and normal modes if , , and :

Compute the generalized eigenvalues of with respect to :

The eigenfrequencies are the square roots of the eigenvalues:

The shapes of the modes are derived from the generalized eigenvectors:

Construct the normal mode solutions as a generalized eigenvector times the corresponding exponential:

Verify that both satisfy the differential equation for the system: