Additional functionality related to this tutorial has been introduced in subsequent versions of the Wolfram Language. For the latest information, see Matrices and Linear Algebra.

Matrix Computations

Basic Operations	Eigensystem Computations
Solving Linear Systems	Matrix Decompositions
Least Squares Solutions	Functions of Matrices

This tutorial reviews the functions that Wolfram Language provides for carrying out matrix computations. Further information on these functions can be found in standard mathematical texts by such authors as Golub and van Loan or Meyer. The operations described in this tutorial are unique to matrices; an exception is the computation of norms, which also extends to scalars and vectors.

Basic Operations

This section gives a review of some basic concepts and operations that will be used throughout the tutorial to discuss matrix operations.

Norms

The norm of a mathematical object is a measurement of the length, size, or extent of the object. In Wolfram Language norms are available for scalars, vectors, and matrices.

Norm[num]	norm of a number
Norm[vec]	2‐norm of a vector
Norm[vec,p]	p‐norm of a vector
Norm[mat]	2‐norm of a matrix
Norm[mat,p]	p‐norm of a matrix
Norm[mat,"Frobenius"]	Frobenius norm of a matrix

Computing norms in Wolfram Language.

For numbers, the norm is the absolute value:

Vector Norms

For vector spaces, norms allow a measure of distance. This allows the definition of many familiar concepts such as neighborhood, closeness of approximation, and goodness of fit. A vector norm is a function that satisfies the following relations.

Typically this function uses the notation . The subscript is used to distinguish different norms, of which the p-norms are particularly important. For , the p-norm is defined as follows.

Some common p-norms are the 1-, 2-, and -norms.

In Wolfram Language, vector p-norms can be computed with the function Norm. The 1-, 2-, and

-norms are demonstrated in the following examples:

The 2-norm is particularly useful and this is the default:

Norms are implemented for vectors with exact numerical entries:

Norms are also implemented for vectors with symbolic entries:

In addition, if a symbolic p is used, the result is symbolic:

Matrix Norms

Matrix norms are used to give a measure of distance in matrix spaces. This is necessary if you want to quantify what it means for one matrix to be near to another, for example, to say that a matrix is nearly singular.

Matrix norms also use the double bar notation of vector norms. One of the most common matrix norms is the Frobenius norm (also called the Euclidean norm).

Other common norms are the p-norms. These are defined in terms of vector p-norms as follows.

Thus the matrix p-norms show the maximum expansion that a matrix can apply to any vector.

In Wolfram Language the Frobenius norm can be computed as follows:

The matrix 2-norm can be computed as follows:

It is also possible to give an argument to specify the 1, 2, or

matrix p-norms:

This computes the expansion that a collection of different vectors gets from an input matrix. You can see that the maximum value is still less than the 2-norm:

All the p-norms are supported for exact numerical matrices:

However, only the 1 and

matrix p-norms are supported for symbolic matrices:

NullSpace

One of the fundamental subspaces associated with each matrix is the null space. Vectors in the null space of a matrix are mapped to zero by the action of the matrix.

NullSpace[m]

a list of basis vectors whose linear combinations satisfy

For some matrices (for instance, nonsingular square matrices), the null space is empty:

This matrix has a null space with one vector:

This matrix has a null space with two basis vectors:

Rank

The rank of a matrix corresponds to the number of linearly independent rows or columns in the matrix.

MatrixRank[mat]

give the rank of the matrix mat

If an m×n matrix has no linear dependencies between its rows, its rank is equal to

and it is said to be full rank:

If a matrix has any dependencies in its rows then its rank is less than

. In this case the matrix is said to be rank deficient:

Note that the rank of a matrix is equal to the rank of its transpose. This means that the number of linearly independent rows is equal to the number of linearly independent columns:

For an m×n matrix A the following relations hold: Length[NullSpace[A]]+ MatrixRank[A]n, and Length[NullSpace[A^T]]+ MatrixRank[A^T]m. From this it follows that the null space is empty if and only if the rank is equal to n and that the null space of the transpose of A is empty if and only if the rank of A is equal to m.

If the rank is equal to the number of columns, it is said to have full column rank. If the rank is equal to the number of rows, it is said to have full row rank. One way to understand the rank of a matrix is to consider the row echelon form.

Reduced Row Echelon Form

A matrix can be reduced to a row echelon form by a combination of row operations that start with a pivot position at the top-left element and subtract multiples of the pivot row from following rows so that all entries in the column below the pivot are zero. The next pivot is chosen by going to the next row and column. If this pivot is zero and any nonzero entries are in the column beneath, the rows are exchanged and the process is repeated. The process finishes when the last row or column is reached.

A matrix is in row echelon form if any row that consists entirely of zeros is followed only by other zero rows and if the first nonzero entry in row is in the column , then elements in columns from 1 to in all rows below are zero. The row echelon form is not unique but its form (in the sense of the positions of the pivots) is unique. It also gives a way to determine the rank of a matrix as the number of nonzero rows.

A matrix is in reduced row echelon form if it is in row echelon form, each pivot is one, and all the entries above (and below) each pivot are zero. It can be formed by a procedure similar to the procedure for the row echelon form, also taking steps to reduce the pivot to one (by division) and reduce entries in the column above each pivot to zero (by subtracting multiples of the current pivot row). The reduced row echelon form of a matrix is unique.

The reduced row echelon form (and row echelon form) give a way to determine the rank of a matrix as the number of nonzero rows. In Wolfram Language the reduced row echelon form of a matrix can be computed by the function RowReduce.

RowReduce[mat]

give the reduced row echelon form of the matrix mat

The reduced row echelon form of this matrix only has one nonzero row. This means that the rank is 1:

This is a 3×2 random matrix whose columns are linearly independent:

The reduced row echelon form has two nonzero rows; thus, its rank should be 2:

MatrixRank also computes the rank as 2:

The reduced row echelon form of the transpose of the matrix also has two nonzero rows. This is consistent with the fact that the rank of a matrix is equal to the rank of its transpose, even when the matrix is rectangular:

Inverse

The inverse of a square matrix A is defined by

where is the identity matrix. The inverse can be computed in Wolfram Language with the function Inverse.

Inverse[mat]

give the inverse of the matrix mat

Here is a sample 2×2 matrix:

This demonstrates that the matrix satisfies the definition:

Not all matrices have an inverse; if a matrix does not have an inverse it is said to be singular. If a matrix is rank-deficient then it is singular:

The matrix inverse can in principle be used to solve the matrix equation

by multiplying it by the inverse of .

However, it is always better to solve the matrix equation directly. Such techniques are discussed in the section "Solving Linear Systems".

PseudoInverse

When the matrix is singular or is not square it is still possible to find an approximate inverse that minimizes . When the 2-norm is used this will find a least squares solution known as the pseudoinverse.

PseudoInverse[mat]

give the pseudoinverse of the matrix mat

This finds the pseudoinverse for the singular matrix defined earlier:

The result is not the identity matrix, but it is "close" to the identity matrix:

This computes the pseudoinverse of a rectangular matrix:

The result is quite close to the identity matrix:

The solution found by the pseudoinverse is a least squares solution. These are discussed in more detail under "Least Squares Solutions".

Determinant

The determinant of an n×n matrix is defined as follows.

It can be computed in Wolfram Language with the function Det.

Det[mat]	determinant of the matrix mat
Minors[mat]	matrix of minors of mat
Minors[mat,k]	k ^th minors of mat

Here is a sample 2×2 matrix:

One useful property of the determinant is that

if and only if

is singular:

As already pointed out, if the matrix is singular, it does not have full rank:

Minors

A minor of a matrix is the determinant of any k×k submatrix. This example uses the function Minors to compute all the 2×2 minors:

The size of the submatrices can be controlled by a second option. Here the 1×1 minors are computed; these are just the entries of the matrix:

Note that Minors[m] is equivalent to Minors[m,n-1]. In general, Minors[m,k] computes determinants of all the possible k×k submatrices that can be generated from the matrix m (this is done by picking different k rows and k columns). For the 2×2 minors of a 4×4 matrix you obtain a 6×6 matrix because there are only 6 different ways to pick 2 rows from 4 rows, and to pick 2 columns from 4 columns:

One of the properties of the rank of a matrix is that it is equal to the size of the largest nonsingular submatrix. This can be demonstrated with Minors. In this example, the following matrix has a rank of 2:

The determinant of the matrix is zero:

When the 2×2 minors are computed you can see that not all are zero, confirming that the rank of the matrix is 2:

Solving Linear Systems

One of the most important uses of matrices is to represent and solve linear systems. This section discusses how to solve linear systems with Wolfram Language. It makes strong use of LinearSolve, the main function provided for this purpose.

Solving a linear system involves solving a matrix equation . Because is an × matrix this is a set of linear equations in unknowns.

When the system is said to be square. If there are more equations than unknowns and the system is overdetermined. If there are fewer equations than unknowns and the system is underdetermined.

square	m=n; solution may or may not exist
overdetermined	m>n; solutions may or may not exist
underdetermined	m<n; no solutions or infinitely many solutions exist
nonsingular	number of independent equations equal to the number of variables and determinant nonzero; a unique solution exists
consistent	at least one solution exists
inconsistent	no solutions exist

Classes of linear systems represented by rectangular matrices.

Note that even though you could solve the matrix equation by computing the inverse via , this is not a recommended method. You should use a function that is designed to solve linear systems directly and in Wolfram Language this is provided by LinearSolve.

LinearSolve[m,b]	a vector x that solves the matrix equation m.x==b
LinearSolve[m]	a function that can be applied repeatedly to many b

Solving linear systems with LinearSolve.

This uses LinearSolve to solve a matrix equation:

The solution can be tested to demonstrate that it solves the problem:

You can always generate the linear equations from the input matrices:

Solve can then be used to find the solution:

LinearSolve also works when the right-hand side of the matrix equation is also a matrix:

If the system is overdetermined, LinearSolve will find a solution if it exists:

If no solution can be found, the system is inconsistent. In this case it might still be useful to find a best-fit solution. This is often done with a least squares technique, which is explored under "Least Squares Solutions".

If the system is underdetermined, there is either no solution or infinitely many solutions and LinearSolve will return one:

A system is consistent if and only if the rank of the matrix

equals the rank of the matrix formed by adding

as an extra column to

(known as the augmented matrix). The following demonstrates that the previous system is consistent:

LinearSolve can work with all the different types of matrices that Wolfram Language supports, including symbolic and exact, machine number, and arbitrary precision. It also works with both dense and sparse matrices. When a sparse matrix is used in LinearSolve, special techniques suitable for sparse matrices are used and the result is the solution vector:

option name	default value
Method	Automatic	the method to use for solving
Modulus	0	take the equation to be modulo n
ZeroTest	(#0&)	the function to be used by symbolic methods for testing for zero

Options for LinearSolve.

LinearSolve has three options that allow users more control. The Modulus and ZeroTest options are used by symbolic techniques and are discussed in the section "Symbolic and Exact Matrices". The Method option allows you to choose methods that are known to be suitable for a particular problem; the default setting of Automatic means that LinearSolve chooses a method based on the input.

Singular Matrices

A matrix is singular if its inverse does not exist. One way to test for singularity is to compute a determinant; this will be zero if the matrix is singular. For example, the following matrix is singular:

For many right-hand sides there is no solution:

However, for certain values there will be a solution:

For the first example, the rank of

does not equal that of the augmented matrix. This confirms that the system is not consistent and cannot be solved by LinearSolve:

For the second example, the rank of

does equal that of the augmented matrix. This confirms that the system is consistent and can be solved by LinearSolve:

In those cases where a solution cannot be found, it is still possible to find a solution that makes a best fit to the problem. One important class of best-fit solutions involves least squares solutions and is discussed in "Least Squares Solutions".

Homogeneous Equations

The homogeneous matrix equation involves a zero right-hand side.

This equation has a nonzero solution if the matrix is singular. To test if a matrix is singular, you can compute the determinant.

To find the solution of the homogeneous equation you can use the function NullSpace. This returns a set of orthogonal vectors, each of which solves the homogeneous equation. In the next example, there is only one vector:

This demonstrates that the solution in fact solves the homogeneous equation:

The function Chop can be used to replace approximate numbers close to 0:

The solution to the homogeneous equation can be used to form an infinite number of solutions to the inhomogeneous equation. This solves an inhomogeneous equation:

The solution does in fact solve the equation:

If you add to sol an arbitrary factor times the homogeneous solution, this new vector also solves the matrix equation:

Estimating and Calculating Accuracy

An important way to quantify the accuracy of the solution of a linear system is to compute the condition number . This is defined as follows for a suitable choice of norm.

It can be shown that for the matrix equation the relative error in is times the relative error in and . Thus, the condition number quantifies the accuracy of the solution. If the condition number of a matrix is large, the matrix is said to be ill-conditioned. You cannot expect a good solution from an ill-conditioned system. For some extremely ill-conditioned systems it is not possible to obtain any solution.

In Wolfram Language an approximation of the condition number can be computed with the function LinearAlgebra`MatrixConditionNumber.

LinearAlgebra`MatrixConditionNumber[mat]
	infinity‐norm approximate condition number of a matrix of approximate numbers
LinearAlgebra`MatrixConditionNumber[mat,Norm->p]
	p‐norm approximate condition number of a matrix, where p may be 1, 2, or

This computes the condition number of a matrix:

This matrix is singular and the condition number is

This matrix has a large condition number and is said to be ill-conditioned:

If a matrix is multiplied with itself, the condition number increases:

When you solve a matrix equation involving an ill-conditioned matrix, the result may not be as accurate:

The solution is less accurate for the matrix mat2:

The solution to these problems is to avoid constructing matrices that may be ill-conditioned, for example, by avoiding multiplying matrices by themselves. Another method of solving problems more accurately is to use Wolfram Language's arbitrary-precision computation; this is discussed under "Arbitrary-Precision Matrices":

Symbolic and Exact Matrices

LinearSolve works for all the different types of matrices that can be represented in Wolfram Language. These are described in more detail in "Matrix Types".

Here is an example of working with purely symbolic matrices:

The original matrix can be multiplied into the solution:

The result of multiplication needs some postprocessing with Simplify to reach the expected value. This demonstrates the need for intermediate processing in symbolic linear algebra computation. Without care these intermediate results can become extremely large and make the computation run slowly:

In order to simplify the intermediate expressions, the ZeroTest option might be useful.

LinearSolve can also be used to solve exact problems:

There are a number of methods that are specific to symbolic and exact computation: CofactorExpansion, DivisionFreeRowReduction, and OneStepRowReduction. These are discussed in the section "Symbolic Methods".

Row Reduction

Row reduction involves adding multiples of rows together so as to produce zero elements when possible. The final matrix is in reduced row echelon form as described previously. If the input matrix is a nonsingular square matrix, the result will be an identity matrix:

This can be used as a way to solve a system of equations:

This can be compared with the following use of LinearSolve:

Saving the Factorization

Many applications of linear systems involve the same matrix

but different right-hand sides

. Because a significant part of the effort involved in solving the system involves processing

, it is common to save the factorization and use it to solve repeated problems. In Wolfram Language, this is done by using a one-argument form of LinearSolve; this returns a functional that you can apply to different vectors to obtain each solution:

When you can apply the LinearSolveFunction to a particular right-hand side the solution is obtained:

This solves the matrix equation:

A different right-hand side yields a different solution:

This new solution solves this matrix equation:

The one-argument form of LinearSolve works in a completely equivalent way to the two-argument form. It works with the same range of input matrices, for example, returning the expected results for symbolic, exact, or sparse matrix input. It also accepts the same options.

One issue with the one-argument form is that for certain input matrices the factorization cannot be saved. For example, if the system is overdetermined, the exact solution is not certain to exist. In this case a message is generated that warns you that the factorization will be repeated each time the functional is applied to a particular right-hand side:

For this right-hand side there is a solution, and this is returned:

However, for this right-hand side there is no solution, and an error is encountered:

Methods

LinearSolve provides a number of different techniques to solve matrix equations 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 Wolfram Language provides for solving matrix equations.

The default setting of the option Method is Automatic. As is typical for Wolfram Language, this means that the system will make an automatic choice of method to use. For LinearSolve if the input matrix is numerical and dense, then a method using LAPACK routines is used; if it is numerical and sparse, a multifrontal direct solver is used. If the matrix is symbolic then specialized symbolic routines are used.

The details of the different methods are now described.

LAPACK

LAPACK is the default method for solving dense numerical matrices. When the matrix is square and nonsingular the routines dgesv, dlange, and dgecon are used for real matrices and zgesv, zlange, and zgecon for complex matrices. When the matrix is nonsquare or singular dgelss is used for real matrices and zgelss for complex matrices. More information on LAPACK is available in the references.

If the input matrix uses arbitrary-precision numbers, then LAPACK algorithms extended for arbitrary-precision computation are used.

Multifrontal

The multifrontal method is a direct solver used by default if the input matrix is sparse; it can also be selected by specifying a method option.

This reads in a sample sparse matrix stored in the Matrix Market format. More information on the importing of matrices is under "Import and Export of Sparse Matrices":

This solves the matrix equation using the sample matrix:

If the input matrix to the multifrontal method is dense, it is converted to a sparse matrix.

The implementation of the multifrontal method uses the "UMFPACK" library.

Krylov

The Krylov method is an iterative solver that is suitable for large sparse linear systems, such as those arising from numerical solving of PDEs. In Wolfram Language two Krylov methods are implemented: conjugate gradient (for symmetric positive definite matrices) and BiCGSTAB (for nonsymmetric systems). It is possible to set the method that is used and a number of other parameters by using appropriate suboptions.

option name	default value
MaxIterations	Automatic	the maximum number of iterations
Method	BiCGSTAB	the method to use for solving
Preconditioner	None	the preconditioner
ResidualNormFunction	Automatic	the norm to minimize
Tolerance	Automatic	the tolerance to use to terminate the iteration

Suboptions of the Krylov method.

The default method for Krylov, BiCGSTAB, is more expensive but more generally applicable. The conjugate gradient method is suitable for symmetric positive definite systems, always converging to a solution (though the convergence may be slow). If the matrix is not symmetric positive definite, the conjugate gradient may not converge to a solution.

In this example, the matrix is not symmetric positive definite and the conjugate gradient method does not converge:

The default method, BiCGSTAB, converges and returns the solution:

This tests the solution that was found:

Here is an example that shows the benefit of the Krylov method and the use of a preconditioner. First, a function that builds a structured banded sparse matrix is defined:

This demonstrates the structure of the matrices generated by this function:

This builds a larger system, with a 10000×10000 matrix:

This will use the default sparse solver, a multifrontal direct solver:

The Krylov method is faster:

The use of the ILU preconditioner is even faster. Currently, the preconditioner can only work if the matrix has nonzero diagonal elements:

At present, only the ILU preconditioner is built-in. You can still define your own preconditioner by defining a function that is applied to the input matrix. An example that involves solving a diagonal matrix is shown next.

First the problem is set up and solved without a preconditioner:

Now, a preconditioner function fun is used; there is a significant improvement to the speed of the computation:

Generally, a problem will not be structured so that it can benefit so much from such a simple preconditioner. However, this example is useful since it shows how to create and use your own preconditioner.

If the input matrix to the Krylov method is dense, the result is still found because the method is based on matrix/vector multiplication.

The Krylov method can be used to solve systems that are too large for a direct solver. However, it is not a general solver, being particularly suitable for those problems that have some form of diagonal dominance.

Cholesky

The Cholesky method is suitable for solving symmetric positive definite systems:

The Cholesky method is faster for this matrix:

The Cholesky method is also more stable. This can be demonstrated with the following matrix:

For this matrix the default LU decomposition method reports a potential error:

The Cholesky method succeeds:

For dense matrices the Cholesky method uses LAPACK functions such as dpotrf and dpotrs for real matrices and zpotrf and zpotrs for complex matrices. For sparse matrices the Cholesky method uses the "TAUCS" library.

Symbolic Methods

There are a number of methods that are specific to symbolic and exact computation: CofactorExpansion, DivisionFreeRowReduction, and OneStepRowReduction. These can be demonstrated for the following symbolic matrix:

The computation is repeated for all three methods:

Least Squares Solutions

This section follows from the previous section. It is concerned with finding solutions to the matrix equation

, where

is an

matrix and

(i.e., when there are more equations than unknowns). In this case, there may well be no solution, and LinearSolve will issue an error message:

In these cases it is possible to find a best-fit solution that minimizes . A particularly common choice of p is 2; this generates a least squares solution. These are often used because the function is differentiable in and because the 2-norm is preserved under orthogonal transformations.

If the rank of is (i.e., it has full column rank), it can be shown (Golub and van Loan) that there is a unique solution to the least squares problem and it solves the linear system.

These are called the normal equations. Although they can be solved directly to get a least squares solution, this is not recommended because if the matrix is ill-conditioned then the product will be even more ill-conditioned. Ways of finding least squares solutions for full rank matrices are explored in "Examples: Full Rank Least Squares Solutions".

It should be noted that best-fit solutions that minimize other norms, such as 1 and , are also possible. A number of ways to find solutions to overdetermined systems that are suitable for special types of matrices or to minimize other norms are given under "Minimization of 1 and Infinity Norms".

A general way to find a least squares solution to an overdetermined system is to use a singular value decomposition to form a matrix that is known as the pseudoinverse of a matrix. In Wolfram Language this is computed with PseudoInverse. This technique works even if the input matrix is rank deficient. The basis of the technique follows.

Here is an overdetermined system; you can see that the matrix

is rank deficient:

The least squares solution can still be found using PseudoInverse:

This shows how close the solution actually is:

Data Fitting

Scientific measurements often result in collections of ordered pairs of data, . In order to make predictions at times other than those that were measured, it is common to try to fit a curve through the data points. If the curve is a straight line, then the aim is to find unknown coefficients and for which

for all the data pairs. This can be written in a matrix form as shown here and is immediately recognized as being equivalent to solving an overdetermined system of linear equations.

Fitting a more general curve, for example,

is equivalent to the matrix equation

In this case the left-hand matrix is a Vandermonde matrix. In fact any function that is linear in the unknown coefficients can be used.

An example of how to find best-fit parameters will now be given. This will start with the measurements shown next:

These can be plotted graphically:

You could try to solve this with the PseudoInverse technique introduced in "PseudoInverse". First, split the data into

and

components:

Now form the left-hand side matrix:

The best-fit parameter can be found as follows:

Wolfram Language provides the function FindFit that can find best-fit solutions to data. The solution agrees with that found by the PseudoInverse:

Finally, a plot is made that shows the data points and the fitted curve:

The function FindFit is quite general; it can also fit functions to data that is not linear in the parameters.

Eigensystem Computations

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 nonsingular, 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.

Wolfram Language has various functions for computing eigenvalues and eigenvectors.

Eigenvalues[m]	a list of the eigenvalues of m
Eigenvectors[m]	a list of the eigenvectors of m
Eigensystem[m]	a list of the form {eigenvalues,eigenvectors}
CharacteristicPolynomial[m,x]	the characteristic polynomial of m

Eigenvalues and eigenvectors.

Here is a sample 2×2 matrix:

This computes the eigenvalues:

This computes the eigenvectors:

You can compute both the eigenvalues and eigenvectors with Eigensystem:

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

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:

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

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

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

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

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

Eigenvalues[m,k]	the largest k eigenvalues of m
Eigenvectors[m,k]	the corresponding eigenvectors of m
Eigenvalues[m,-k]	the smallest k eigenvalues of m
Eigenvectors[m,-k]	the corresponding eigenvectors of m

This is particularly useful if the matrix is very large. Here is a 10000×10000 sparse matrix:

This is the largest eigenvalue:

option name	default value
Cubics	False	whether to return radicals for 3×3 matrices
Method	Automatic	the method to use for solving
Quartics	False	whether to return radicals for 4×4 matrices
ZeroTest	(#0&)	the function to be used by symbolic methods for testing for zero

Options for Eigensystem.

Eigensystem has four options that allow users more control. The Cubics, Quartics, and ZeroTest options are used by symbolic techniques and are discussed under "Symbolic and Exact Matrices". The Method option allows knowledgeable users to choose methods that are particularly appropriate for their problems; the default setting of Automatic means that Eigensystem makes a choice of a method that is generally suitable.

Eigensystem Properties

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 ^th degree polynomial, there will always be eigenvalues.

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

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:

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:

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

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

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

The eigenvectors are an orthonormal basis:

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

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

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:

The eigenvectors are an orthonormal basis:

A Hermitian matrix that has positive eigenvalues is known as positive definite:

Diagonalizing a Matrix

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

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:

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