The Wolfram Language's matrix operations handle both numeric and symbolic matrices, automatically accessing large numbers of highly efficient algorithms. The Wolfram Language uses state-of-the-art algorithms to work with both dense and sparse matrices, and incorporates a number of powerful original algorithms, especially for high-precision and symbolic matrices.

+,*,^,...— all automatically work element-wise

Dot (.)— products of matrices, automatically handling row and column vectors

Inverse— matrix inverse (useLinearSolve for linear systems)

MatrixRank— rank of a matrix

NullSpace— vectors spanning the null space of a matrix

RangeSpace— vectors spanning the range space of a matrix

RowReduce— reduced row echelon form

PseudoInverse— pseudoinverse of a square or rectangular matrix

Transpose— transpose (, entered withtr)

ConjugateTranspose— conjugate transpose (, entered withct)

LowerTriangularize,UpperTriangularize— extract the lower- or upper-triangular part of a matrix

Symmetrize— find the symmetric, antisymmetric, etc. part of a matrix

Diagonal— get the list of elements on the diagonal

Tr— trace

Det— determinant

Norm— operator norm,p-norms and Frobenius norm

Adjugate— adjugate

Minors— matrices of minors

Permanent— permanent

KroneckerProduct— matrix direct product (outer product)

MatrixPower— powers of numeric or symbolic matrices

MatrixExp— matrix exponential

MatrixLog— matrix logarithm

MatrixFunction— general matrix function

Eigenvalues,Eigenvectors— exact or approximate eigenvalues and eigenvectors

Eigensystem— eigenvalues and eigenvectors together

CharacteristicPolynomial— symbolic characteristic polynomial

MatrixMinimalPolynomial

Related Tech Notes

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• Vectors and Matrices
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• Basic Matrix Operations
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• Eigenvalues and Eigenvectors
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• Tensors

Related Guides

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• Linear Systems
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• Matrix-Based Minimization
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• Matrix Decompositions
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• Parts of Matrices
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• Matrix Distributions

Related Links

• Demonstrations related to Matrix Operations (Wolfram Demonstrations Project)
• Wolfram Core Areas: Calculus & Algebra