The solution of the eigenvalue problem is one of the major areas for matrix computations. It has many applications in physics, chemistry, and engineering. For an matrix the eigenvalues are the roots of its characteristic polynomial, . The set of roots, , are called the spectrum of the matrix. For each eigenvalue, , the vectors, , that satisfy

are described as eigenvectors. The matrix of the eigenvectors, , if it exists and is non-singular, may be used as a similarity transformation to form a diagonal matrix with the eigenvalues on the diagonal elements. Many important applications of eigenvalues involve the diagonalization of matrices in this way.

Mathematica has various functions for computing eigenvalues and eigenvectors.

Here is a sample 22 matrix.

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This computes the eigenvalues.

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This computes the eigenvectors.

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You can compute both the eigenvalues and eigenvectors with Eigensystem.

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You can confirm that the first eigenvalue and its eigenvector satisfy the definition.

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You should note that Eigenvectors and Eigensystem return a list of eigenvectors. This means that if you want a matrix with columns that are the eigenvectors you must take a transpose. Here, the eigenvalues and their corresponding eigenvectors are shown to meet the definition of the eigenvectors.

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You can also obtain the characteristic polynomial of the test matrix.

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This finds the roots of the characteristic polynomial; they are seen to match the eigenvalues of the matrix.

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Eigenvalue computations work for sparse matrices as well as for dense matrices. In the following example, one of the eigenvalues is zero.

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An eigenvalue being zero means that the null space of the matrix is not empty.

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Certain applications of eigenvalues do not require all of the eigenvalues to be computed. Mathematica provides a mechanism for obtaining only some eigenvalues.

This is particularly useful if the matrix is very large. Here is a 1000010000 sparse matrix.

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This is the largest eigenvalue.

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This section describes certain properties of eigensystem computations. It should be remembered that because the eigenvalues of an matrix can be associated with the roots of an degree polynomial, there will always be eigenvalues.

Even if a matrix only contains real numbers, its eigenvalues may not be real.

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The eigenvalues of a matrix may be repeated; eigenvalues that are not repeated are called simple. In this example, the repeated eigenvalue has as many eigenvectors as the number of times it is repeated; this is a semisimple eigenvalue.

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The repeated eigenvalue may have fewer eigenvectors than the number of times it is repeated. In this case it is referred to as a defective eigenvalue and the matrix is referred to as a defective matrix.

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A matrix may have a zero eigenvalue, but can still have a complete set of eigenvectors.

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A matrix may have a zero eigenvalue, but not have a complete set of eigenvectors.

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If a matrix is real and symmetric all of the eigenvalues are real, and there is an orthonormal basis of eigenvectors.

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The eigenvectors are an orthonormal basis.

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A symmetric real matrix that has positive eigenvalues is known as positive definite.

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A symmetric real matrix that has nonzero eigenvalues is known as positive semidefinite.

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If a matrix is complex and hermitian (i.e. equal to its conjugate transpose), all of the eigenvalues are real and there is an orthonormal basis of eigenvectors.

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The eigenvectors are an orthonormal basis.

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A hermitian matrix that has positive eigenvalues is known as positive definite.

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One of the uses of eigensystem computations is to provide a similarity transformation that diagonalizes a matrix.

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Similarity transformations preserve a number of useful properties of a matrix such as determinant, rank, and trace.

If the matrix of eigenvectors is singular then the matrix cannot be diagonalized. This happens if the matrix is defective (i.e., there is an eigenvalue that does not have a complete set of eigenvectors) as in this example.

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The matrix of column eigenvectors is singular and so a similarity transformation that diagonalizes the original matrix does not exist.

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As is typical for Mathematica, if a computation is done for a symbolic or exact matrix it will use symbolic computer algebra techniques and return a symbolic or exact result.

This is a sample 33 matrix of integers.

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This computes the eigenvalues; the result uses Root objects.

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A machine-precision approximation to the result can be computed by using N.

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The result can be returned in terms of radicals by setting the Cubics option to True.

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For nn matrices the generalized eigenvalues are the n roots of its characteristic polynomial, . For each generalized eigenvalue, , the vectors, , that satisfy

are described as generalized eigenvectors.

Here are two 22 matrices.

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This computes the generalized eigenvalues.

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If the two matrices share a common nonzero nullspace, there will be an indeterminate eigenvalue.

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These two matrices share a common one-dimensional nullspace.

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One of the generalized eigenvalues is Indeterminate.

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Eigenvalue computations are solved with a number of different techniques that are specialized to particular problems. You can select between these using the option Method. In this way a uniform interface is provided to all the functionality that Mathematica provides.

Here is a 33 sparse matrix.

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This computes the largest eigenvalue, specifying that the Arnoldi method is used.

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This computes the eigenvalue near to -1.

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If a shift is given that matches an eigenvalue an error results.

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This tries to find eigenvalues of a 15001500 tridiagonal matrix with 2 on the diagonal and -1 off the diagonal. The algorithm does not converge.

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When the technique does not converge it returns a result, but this may not be very accurate. One way to accelerate the convergence would be to use the results generated as a shift.

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An alternative way to get convergence is to increase the number of iterations.

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Alternatively, the convergence may be achieved by increasing the Arnoldi basis size in addition to the number of iterations. In this example, the increased basis size makes the algorithm converge in a smaller number of iterations. Even though each iteration is more expensive, the overall computation is faster.

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The Arnoldi algorithm with a basis size of 30 is still faster and less memory intensive than the dense eigenvalue computation.

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For many large sparse systems that occur in practical computations the Arnoldi algorithm is able to converge quite quickly.

Symbolic eigenvalue computations work by interpolating the characteristic polynomial.