Matrices in the Wolfram Language are represented as lists of lists. You can use all the standard Wolfram Language list‐manipulation operations on matrices.

A range of indices can be specified by using ;; (Span).

The Wolfram Language represents matrices and vectors using lists. Anything that is not a list the Wolfram Language considers as a scalar.

A vector in the Wolfram Language consists of a list of scalars. A matrix consists of a list of vectors, representing each of its rows. In order to be a valid matrix, all the rows must be the same length, so that the elements of the matrix effectively form a rectangular array.

Most mathematical functions in the Wolfram Language are set up to apply themselves separately to each element in a list. This is true in particular of all functions that carry the attribute Listable.

A consequence is that most mathematical functions are applied element by element to matrices and vectors.

It is important to realize that the Wolfram Language treats an object as a vector in a particular operation only if the object is explicitly a list at the time when the operation is done. If the object is not explicitly a list, the Wolfram Language always treats it as a scalar. This means that you can get different results, depending on whether you assign a particular object to be a list before or after you do a particular operation.

It is important to realize that you can use "dot" for both left‐ and right‐multiplication of vectors by matrices. The Wolfram Language makes no distinction between "row" and "column" vectors. Dot carries out whatever operation is possible. (In formal terms, contracts the last index of the tensor with the first index of .)

For some purposes, you may need to represent vectors and matrices symbolically without explicitly giving their elements. You can use Dot to represent multiplication of such symbolic objects.

The "dot" operator gives "inner products" of vectors, matrices, and so on. In more advanced calculations, you may also need to construct outer or Kronecker products of vectors and matrices. You can use the general function Outer or KroneckerProduct to do this.

Outer products are discussed in more detail in "Tensors".

Two vectors are orthogonal if their dot product is zero. A set of vectors is orthonormal if they are all unit vectors and are pairwise orthogonal.

If you give a matrix with exact symbolic or numerical entries, the Wolfram Language gives the exact inverse. If, on the other hand, some of the entries in your matrix are approximate real numbers, then the Wolfram Language finds an approximate numerical result.

When you try to invert a matrix with exact numerical entries, the Wolfram Language can always tell whether or not the matrix is singular. When you invert an approximate numerical matrix, The Wolfram Language can usually not tell for certain whether or not the matrix is singular: all it can tell is, for example, that the determinant is small compared to the entries of the matrix. When the Wolfram Language suspects that you are trying to invert a singular numerical matrix, it prints a warning.

If you work with high‐precision approximate numbers, the Wolfram Language will keep track of the precision of matrix inverses that you generate.

Inverse works only on square matrices. "Advanced Matrix Operations" discusses the function PseudoInverse, which can also be used with nonsquare matrices.

Transposing a matrix interchanges the rows and columns in the matrix. If you transpose an m×n matrix, you get an n×m matrix as the result.

Det[m] gives the determinant of a square matrix m. Minors[m] is the matrix whose ^th element gives the determinant of the submatrix obtained by deleting the ^th row and the ^th column of m. The ^th cofactor of m is times the ^th element of the matrix of minors.

Minors[m,k] gives the determinants of the k×k submatrices obtained by picking each possible set of k rows and k columns from m. Note that you can apply Minors to rectangular, as well as square, matrices.

The trace or spur of a matrix Tr[m] is the sum of the terms on the leading diagonal.

The rank of a matrix is the number of linearly independent rows or columns.

The matrix exponential of a matrix m is , where indicates a matrix power.

Many calculations involve solving systems of linear equations. In many cases, you will find it convenient to write down the equations explicitly, and then solve them using Solve.

In some cases, however, you may prefer to convert the system of linear equations into a matrix equation, and then apply matrix manipulation operations to solve it. This approach is often useful when the system of equations arises as part of a general algorithm, and you do not know in advance how many variables will be involved.

A system of linear equations can be stated in matrix form as , where is the vector of variables.

Note that if your system of equations is sparse, so that most of the entries in the matrix are zero, then it is best to represent the matrix as a SparseArray object. As discussed in "Sparse Arrays: Linear Algebra", you can convert from symbolic equations to SparseArray objects using CoefficientArrays. All the functions described here work on SparseArray objects as well as ordinary matrices.

If you have a square matrix with a nonzero determinant, then you can always find a unique solution to the matrix equation for any . If, however, the matrix has determinant zero, then there may be either no vector, or an infinite number of vectors which satisfy for a particular . This occurs when the linear equations embodied in are not independent.

When has determinant zero, it is nevertheless always possible to find nonzero vectors that satisfy . The set of vectors satisfying this equation form the null space or kernel of the matrix . Any of these vectors can be expressed as a linear combination of a particular set of basis vectors, which can be obtained using NullSpace[m].

NullSpace and MatrixRank have to determine whether particular combinations of matrix elements are zero. For approximate numerical matrices, the Tolerance option can be used to specify how close to zero is considered good enough. For exact symbolic matrices, you may sometimes need to specify something like ZeroTest->(FullSimplify[#]==0&) to force more to be done to test whether symbolic expressions are zero.

An important feature of functions like LinearSolve and NullSpace is that they work with rectangular, as well as square, matrices.

When you represent a system of linear equations by a matrix equation of the form , the number of columns in gives the number of variables, and the number of rows gives the number of equations. There are a number of cases.

The number of independent equations is the rank of the matrix MatrixRank[m]. The number of redundant equations is Length[NullSpace[m]]. Note that the sum of these quantities is always equal to the number of columns in m.

In some applications, you will want to solve equations of the form many times with the same , but different . You can do this efficiently in the Wolfram Language by using LinearSolve[m] to create a single LinearSolveFunction that you can apply to as many vectors as you want.

The eigenvalues of a matrix are the values for which one can find nonzero vectors such that . The eigenvectors are the vectors .

The characteristic polynomial CharacteristicPolynomial[m,x] for an × matrix is given by Det[m-x IdentityMatrix[n]]. The eigenvalues are the roots of this polynomial.

Finding the eigenvalues of an × matrix in general involves solving an ^th‐degree polynomial equation. For , therefore, the results cannot in general be expressed purely in terms of explicit radicals. Root objects can nevertheless always be used, although except for fairly sparse or otherwise simple matrices the expressions obtained are often unmanageably complex.

If you give a matrix of approximate real numbers, the Wolfram Language will find the approximate numerical eigenvalues and eigenvectors.

The function Eigenvalues always gives you a list of eigenvalues for an × matrix. The eigenvalues correspond to the roots of the characteristic polynomial for the matrix, and may not necessarily be distinct. Eigenvectors, on the other hand, gives a list of eigenvectors which are guaranteed to be independent. If the number of such eigenvectors is less than , then Eigenvectors supplements the list with zero vectors, so that the total length of the list is always .

Eigenvalues sorts numeric eigenvalues so that the ones with large absolute value come first. In many situations, you may be interested only in the largest or smallest eigenvalues of a matrix. You can get these efficiently using Eigenvalues[m,k] and Eigenvalues[m,-k].

The generalized eigenvalues for a matrix with respect to a matrix are defined to be those for which .

The generalized eigenvalues correspond to zeros of the generalized characteristic polynomial Det[m-x a].

Note that while ordinary matrix eigenvalues always have definite values, some generalized eigenvalues will always be Indeterminate if the generalized characteristic polynomial vanishes, which happens if and share a null space. Note also that generalized eigenvalues can be infinite.

The singular values of a matrix are the square roots of the eigenvalues of , where denotes Hermitian transpose. The number of such singular values is the smaller dimension of the matrix. SingularValueList sorts the singular values from largest to smallest. Very small singular values are usually numerically meaningless. With the option setting Tolerance->t, SingularValueList drops singular values that are less than a fraction t of the largest singular value. For approximate numerical matrices, the tolerance is by default slightly greater than zero.

If you multiply the vector for each point in a unit sphere in ‐dimensional space by an × matrix , then you get an ‐dimensional ellipsoid, whose principal axes have lengths given by the singular values of .

The 2‐norm of a matrix Norm[m,2] is the largest principal axis of the ellipsoid, equal to the largest singular value of the matrix. This is also the maximum 2‐norm length of for any possible unit vector .

The ‐norm of a matrix Norm[m,p] is in general the maximum ‐norm length of that can be attained. The cases most often considered are , , and . Also sometimes considered is the Frobenius norm Norm[m,"Frobenius"], which is the square root of the trace of .

When you create a LinearSolveFunction using LinearSolve[m], this often works by decomposing the matrix into triangular forms, and sometimes it is useful to be able to get such forms explicitly.

LU decomposition effectively factors any square matrix into a product of lower‐ and upper‐triangular matrices. Cholesky decomposition effectively factors any Hermitian positive‐definite matrix into a product of a lower‐triangular matrix and its Hermitian conjugate, which can be viewed as the analog of finding a square root of a matrix.

The standard definition for the inverse of a matrix fails if the matrix is not square or is singular. The pseudoinverse of a matrix can however still be defined. It is set up to minimize the sum of the squares of all entries in , where is the identity matrix. The pseudoinverse is sometimes known as the generalized inverse, or the Moore–Penrose inverse. It is particularly used for problems related to least‐squares fitting.

QR decomposition writes any matrix as a product , where is an orthonormal matrix, denotes Hermitian transpose, and is a triangular matrix, in which all entries below the leading diagonal are zero.

Singular value decomposition, or SVD, is an underlying element in many numerical matrix algorithms. The basic idea is to write any matrix in the form , where is a matrix with the singular values of on its diagonal, and are orthonormal matrices, and is the Hermitian transpose of .

Most square matrices can be reduced to a diagonal matrix of eigenvalues by applying a matrix of their eigenvectors as a similarity transformation. But even when there are not enough eigenvectors to do this, one can still reduce a matrix to a Jordan form in which there are both eigenvalues and Jordan blocks on the diagonal. Jordan decomposition in general writes any square matrix in the form as .

Numerically more stable is the Schur decomposition, which writes any square matrix in the form , where is an orthonormal matrix, and is block upper‐triangular. Also related is the Hessenberg decomposition, which writes a square matrix in the form , where is an orthonormal matrix, and can have nonzero elements down to the diagonal below the leading diagonal.

Tensors are mathematical objects that give generalizations of vectors and matrices. In the Wolfram System, a tensor is represented as a set of lists, nested to a certain number of levels. The nesting level is the rank of the tensor.

A tensor of rank k is essentially a k‐dimensional table of values. To be a true rank k tensor, it must be possible to arrange the elements in the table in a k‐dimensional cuboidal array. There can be no holes or protrusions in the cuboid.

The indices that specify a particular element in the tensor correspond to the coordinates in the cuboid. The dimensions of the tensor correspond to the side lengths of the cuboid.

One simple way that a rank k tensor can arise is in giving a table of values for a function of k variables. In physics, the tensors that occur typically have indices which run over the possible directions in space or spacetime. Notice, however, that there is no built‐in notion of covariant and contravariant tensor indices in the Wolfram System: you have to set these up explicitly using metric tensors.

The rank of a tensor is equal to the number of indices needed to specify each element. You can pick out subtensors by using a smaller number of indices.

You can think of a rank k tensor as having k "slots" into which you insert indices. Applying Transpose is effectively a way of reordering these slots. If you think of the elements of a tensor as forming a k‐dimensional cuboid, you can view Transpose as effectively rotating (and possibly reflecting) the cuboid.

In the most general case, Transpose allows you to specify an arbitrary reordering to apply to the indices of a tensor. The function Transpose[T,{p₁,p₂,…,p_k}] gives you a new tensor T^′ such that the value of T^′_{i1 i2 … ik} is given by T_{ip1 ip2 … ipk}.

If you originally had an n_p1×n_p2×…×n_pk tensor, then by applying Transpose, you will get an n₁×n₂×…×n_k tensor.

If you have a tensor that contains lists of the same length at different levels, then you can use Transpose to effectively collapse different levels.

You can also use Tr to extract diagonal elements of a tensor.

Outer products, and their generalizations, are a way of building higher‐rank tensors from lower‐rank ones. Outer products are also sometimes known as direct, tensor, or Kronecker products.

From a structural point of view, the tensor you get from Outer[f,t,u] has a copy of the structure of u inserted at the "position" of each element in t. The elements in the resulting structure are obtained by combining elements of t and u using the function f.

If you take the generalized outer product of an m₁×m₂×…×m_r tensor and an n₁×n₂×…×n_s tensor, you get an m₁×…×m_r×n₁×…×n_s tensor. If the original tensors have ranks r and s, your result will be a rank r+s tensor.

In terms of indices, the result of applying Outer to two tensors T_{i1 i2 … ir} and U_{j1 j2 … js} is the tensor V_{i1 i2 … irj1 j2 … js} with elements f[T_{i1 i2 … ir},U_{j1 j2 … js}].

In doing standard tensor calculations, the most common function f to use in Outer is Times, corresponding to the standard outer product.

Particularly in doing combinatorial calculations, however, it is often convenient to take f to be List. Using Outer, you can then get combinations of all possible elements in one tensor, with all possible elements in the other.

In constructing Outer[f,t,u] you effectively insert a copy of u at every point in t. To form Inner[f,t,u], you effectively combine and collapse the last dimension of t and the first dimension of u. The idea is to take an m₁×m₂×…×m_r tensor and an n₁×n₂×…×n_s tensor, with m_r=n₁, and get an m₁×m₂×…×m_r-1×n₂×…×n_s tensor as the result.

The simplest examples are with vectors. If you apply Inner to two vectors of equal length, you get a scalar. Inner[f,v₁,v₂,g] gives a generalization of the usual scalar product, with f playing the role of multiplication, and g playing the role of addition.

You can think of Inner as performing a "contraction" of the last index of one tensor with the first index of another. If you want to perform contractions across other pairs of indices, you can do so by first transposing the appropriate indices into the first or last position, then applying Inner, and then transposing the result back.

In many applications of tensors, you need to insert signs to implement antisymmetry. The function Signature[{i₁,i₂,…}], which gives the signature of a permutation, is often useful for this purpose.

Many large-scale applications of linear algebra involve matrices that have many elements, but comparatively few that are nonzero. You can represent such sparse matrices efficiently in the Wolfram System using SparseArray objects, as discussed in "Sparse Arrays: Manipulating Lists". SparseArray objects work by having lists of rules that specify where nonzero values appear.

As discussed in "Sparse Arrays: Manipulating Lists", you can use patterns to specify collections of elements in sparse arrays. You can also have sparse arrays that correspond to tensors of any rank.

You can apply most standard structural operations directly to SparseArray objects, just as you would to ordinary lists. When the results are sparse, they typically return SparseArray objects.

For machine-precision numerical sparse matrices, the Wolfram System supports standard file formats such as Matrix Market (.mtx) and Harwell–Boeing. You can import and export matrices in these formats using Import and Export.