Documentation

matrix

Create a matrix or a vector

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Syntax

matrix(Array)
matrix(List)
matrix(ListOfRows)
matrix(Matrix)
matrix(m, n)
matrix(m, n, Array)
matrix(m, n, List)
matrix(m, n, ListOfRows)
matrix(m, n, Table)
matrix(m, n, [(i1, j1) = value1, (i2, j2) = value2, …])
matrix(m, n, f)
matrix(m, n, List, Diagonal)
matrix(m, n, g, Diagonal)
matrix(m, n, List, Banded)
matrix(1, n, [j1 = value1, j2 = value2, …])
matrix(m, 1, [i1 = value1, i2 = value2, …])

Description

matrix(m, n, [[a11, a12, ...], [a21, a22, ...], ...]) returns an m×n matrix of the domain type Dom::Matrix().

matrix(m, n, [a11, a12, ..., a21, a22, ..., a.m.n]) returns an m×n matrix of the domain type Dom::Matrix().

matrix(m, 1, [a1, a2, ...]) returns an m×1 column vector of the domain type Dom::Matrix().

matrix(1, n, [a1, a2, ...]) returns an 1 ×n row vector of the domain type Dom::Matrix().

matrix is equivalent to Dom::Matrix().

matrix creates matrices and vectors. A column vector is represented as an m×1 matrix. A row vector is represented as a n matrix.

Matrix and vector components must be arithmetical expressions (numbers and/or symbolic expressions). If matrices over special component rings are desired, use the domain constructor Dom::Matrix with a suitable component ring.

Arithmetical operations with matrices can be performed by using the standard arithmetical operators of MuPAD®.

E.g., if A and B are two matrices defined by matrix, then A + B computes the sum and A * B computes the product of the two matrices, provided that the dimensions are appropriate.

Similarly, A^(-1) or 1/A computes the inverse of a square matrix A if it can be inverted. Otherwise, FAIL is returned.

Cf. Example 1.

Many system functions accept matrices as input, such as map, subs, has, zip, conjugate, norm or exp. Cf. Example 4.

Most of the functions in the MuPAD linear algebra package linalg work with matrices. For example, the command linalg::gaussJordan(A) performs Gauss-Jordan elimination on A to transform A to its reduced row echelon form.

For numerical matrix computations, the corresponding functions of the numeric package accept matrices.

Matrix components can be extracted by the usual index operator [ ], which also works for lists, arrays, and tables. The call A[i, j] extracts the matrix component in the i-th row and the j-th column.

Assignments to matrix components are performed similarly. The call A[i, j] := c replaces the matrix component in the i-th row and the j-th column of A by c.

If one of the indices is not in its valid range, an error message is issued.

The index operator also extracts submatrices. The call A[r1..r2, c1..c2] creates the submatrix of A comprising the rows with the indices r1, r1 + 1, …, r2 and the columns with the indices c1, c1 + 1, …, c2 of A.

See Example 3 and Example 5.

matrix(Array) or matrix(Matrix) create a new matrix with the same dimension and the components of Array or Matrix, respectively. The array must not contain any uninitialized entries. If Array is one-dimensional, the result is a column vector. Cf. Example 8.

matrix(List) creates an m×1 column vector with components taken from the non-empty list, where m is the number of entries of List. Cf. Example 5.

matrix(ListOfRows) creates an m×n matrix with components taken from the nested list ListOfRows, where m is the number of inner lists of ListOfRows, and n is the maximal number of elements of an inner list. Each inner list corresponds to a row of the matrix. Both m and n must be non-zero.

If a row has less than n entries, the remaining entries in the corresponding row of the matrix are regarded as zero. Cf. Example 7.

The call matrix(m, n) returns the m×n zero matrix.

The call matrix(m, n, Array) creates an m×n matrix with components taken from Array, which must be an array or an hfarray. Array must have mn operands. The first m operands define the first row, the next m operands define the second row, etc. The formatting of the array is irrelevant. E.g., any array with 6 elements can be used to create matrices of dimension 1 ×6, or 2×3, or 3×2, or 6 ×1.

matrix(m, n, List) creates an m×n matrix with components taken row after row from the non-empty list. The list must contain mn entries. Cf. Example 7.

matrix(m, n, ListOfRows) creates an m×n matrix with components taken from the list ListOfRows.

If m ≥ 2 and n ≥ 2, then ListOfRows must consist of at most m inner lists, each having at most n entries. The inner lists correspond to the rows of the returned matrix.

If a row has less than n entries, the remaining components of the corresponding row of the matrix are regarded as zero. If there are less than m rows, the remaining lower rows of the matrix are filled with zeroes. Cf. Example 7.

matrix(m,n,Table) creates an m×n matrix with components taken from the table Table. The table entries Table[i,j] with positive integer values of i and j define the corresponding entries of the matrix. Zero entries need not be specified in the table. This way, sparse table input can be used to create the matrix.

For large sparse matrices, the fastest way of creation is the generation of an empty table that is filled by indexed assignments and then passed to matrix. Alternatively, one may first create an empty sparse matrix via matrix(m, n) and then fill in the non-zero entries via indexed assignments. Note that the indexed assignment to a matrix is somewhat slower than the indexed assignment to a table.

matrix(m, n, [(i1, j1) = value1, (i2, j2) = value2, ...]) is a further way to create a matrix specifying only the non-zero entries A[i1, j1] = value1, A[i2, j2] = value2 etc. The ordering of the entries in the input list is irrelevant.

matrix(m, n, f) returns the matrix whose (i, j)-th component is the return value of the function call f(i,j). The row index i runs from 1 to m and the column index j from 1 to n. Cf. Example 9.

matrix(m, 1, Array) returns the m×1 column vector with components taken from Array. The array or hfarray Array must have m entries.

matrix(m, 1, List) returns the m×1 column vector with components taken from List. The list List must have no more than m entries. If there are fewer entries, the remaining vector components are regarded as zero. Cf. Example 5.

matrix(m, 1, Table) returns the m×1 column vector with components taken from Table. The table Table must have no more than m entries. If there are fewer entries, the remaining vector components are regarded as zero. Cf. Example 6.

matrix(m, 1, [i1 = value1, i2 = value2, ...]) provides a way to create a sparse column vector specifying only the non-zero entries A[i1] = value1, A[i2] = value2 etc. The ordering of the entries in the input list is irrelevant.

matrix(1, n, Array) returns the 1 ×n row vector with components taken from Array. The array or hfarray Array must have n entries.

matrix(1, n, List) returns the 1 ×n row vector with components taken from List. The list List must not have more than n entries. If there are fewer entries, the remaining vector components are regarded as zero. Cf. Example 5.

matrix(1, n, Table) returns the 1 ×n row vector with components taken from Table. The table Table must not have more than n entries. If there are fewer entries, the remaining vector components are regarded as zero. Cf. Example 6.

matrix(1, n, [j1 = value1, j2 = value2, ...]) provides a way to create a sparse row vector specifying only the non-zero entries A[j1] = value1, A[j2] = value2 etc. The ordering of the entries in the input list is irrelevant.

Note

The number of rows and columns, respectively, of a matrix must be less than 231.

Note

The components of a matrix are no longer evaluated after the creation of the matrix, i.e., if they contain free identifiers they will not be replaced by their values.

Examples

Example 1

We create a 2×2 matrix by passing a list of two rows to matrix, where each row is a list of two elements:

A := matrix([[1, 5], [2, 3]]) In the same way, we generate the following 2 ×3 matrix:

B := matrix([[-1, 5/2, 3], [1/3, 0, 2/5]]) We can do matrix arithmetic using the standard arithmetical operators of MuPAD. For example, the matrix product AB, the fourth power of A, and the scalar multiplication of A by are given by:

A * B, A^4, 1/3 * A Since the dimensions of the matrices A and B differ, the sum of A and B is not defined and MuPAD returns an error message:

A + B
Error: Dimensions do not match. [(Dom::Matrix(Dom::ExpressionField()))::_plus]

To compute the inverse of A, enter:

1/A If a matrix is not invertible, the result of this operation is FAIL:

C := matrix([[2, 0], [0, 0]]) C^(-1) delete A, B, C:

Example 2

In addition to standard matrix arithmetic, the library linalg offers numerous functions handling matrices. For example, the function linalg::rank determines the rank of a matrix:

A := matrix([[1, 5], [2, 3]]) linalg::rank(A) The function linalg::eigenvectors computes the eigenvalues and the eigenvectors of A:

linalg::eigenvectors(A) To determine the dimension of a matrix, use the function linalg::matdim:

linalg::matdim(A) The result is a list of two positive integers, the row and column number of the matrix.

Use info(linalg) to obtain a list of available functions, or enter ?linalg for details about this library.

delete A:

Example 3

Matrix entries can be accessed with the index operator [ ]:

A := matrix([[1, 2, 3, 4], [2, 0, 4, 1], [-1, 0, 5, 2]]) A[2, 1] * A[1, 2] - A[3, 1] * A[1, 3] You can redefine a matrix entry by assigning a value to it:

A[1, 2] := a^2: A The index operator can also be used to extract submatrices. The following call creates a copy of the submatrix of A comprising the second and the third row and the first three columns of A:

A[2..3, 1..3] The index operator does not allow to replace a submatrix of a given matrix by another matrix. Use linalg::substitute to achieve this.

delete A:

Example 4

Some system functions can be applied to matrices. For example, if you have a matrix with symbolic entries and want to have all entries in expanded form, simply apply the function expand:

delete a, b:
A := matrix([
[(a - b)^2, a^2 + b^2],
[a^2 + b^2, (a - b)*(a + b)]
]) expand(A) You can differentiate all matrix components with respect to some indeterminate:

diff(A, a) The following command evaluates all matrix components at a given point:

subs(A, a = 1, b = -1) Note that the function subs does not evaluate the result of the substitution. For example, we define the following matrix:

A := matrix([[sin(x), x], [x, cos(x)]]) Then we substitute x = 0 in each matrix component:

B := subs(A, x = 0) You see that the matrix components are not evaluated completely. For example, if you enter sin(0) directly, it evaluates to zero.

The function eval can be used to evaluate the result of the function subs. However, eval does not operate on matrices directly, and you must use the function map to apply the function eval to each matrix component:

map(B, eval) The function zip can be applied to matrices. The following call combines two matrices A and B by dividing each component of A by the corresponding component of B:

A := matrix([[4, 2], [9, 3]]):
B := matrix([[2, 1], [3, -1]]):
A, B, zip(A, B, `/`) delete A, B:

Example 5

A vector is either an m×1 matrix (a column vector) or a n matrix (a row vector). To create a vector with matrix, pass the dimension of the vector and a list of vector components as argument to matrix:

row_vector := matrix(1, 3, [1, 2, 3]);
column_vector := matrix(3, 1, [1, 2, 3])  If the only argument of matrix is a non-nested list or a one-dimensional array, the result is a column vector:

matrix([1, 2, 3]) For a row vector r, the calls r[1, i] and r[i] both return the i-th vector component of r. Similarly, for a column vector c, the calls c[i, 1] and c[i] both return the i-th vector component of c.

We extract the second component of the vectors defined above:

row_vector = row_vector[1, 2],
column_vector = column_vector[2, 1] Use the function linalg::vecdim to determine the number of components of a vector:

linalg::vecdim(row_vector), linalg::vecdim(column_vector) The number of components of a vector can also be determined directly by the call nops(vector).

The dimension of a vector can be determined as described above in the case of matrices:

linalg::matdim(row_vector),
linalg::matdim(column_vector) See the linalg package for functions working with vectors, and the help page of norm for computing vector norms.

delete row_vector, column_vector:

Example 6

A vector is either an m×1 matrix (a column vector) or a n matrix (a row vector). To create a vector with matrix, one may also pass the dimension of the vector and a table of vector components as argument to matrix:

delete v1, v2, t1, t2:
t1 := table():
t1[1,1] := 1:
t1[1,2] := 2:
t1[1,3] := 3:
v1 := matrix(1, 3, t1); t2 := table():
t2[1,1] := 1:
t2[2,1] := 2:
t2[3,1] := 3:
v2 := matrix(3, 1, t2); All functions applied to the vectors in the previous example (see above) can can also be used on these vectors.

delete t1, t2, v1, v2:

Example 7

In the following examples, we illustrate various calls of matrix as described above. We start by passing a nested list to matrix, where each inner list corresponds to a row of the matrix:

matrix([[1, 2], ]) The number of rows of the created matrix is the number of inner lists, namely m = 2. The number of columns is determined by the maximal number of entries of an inner list. In the example above, the first list is the longest one, and hence n = 2. The second list has only one element and, therefore, the second entry in the second row of the returned matrix was set to zero.

In the following call, we use the same nested list, but in addition pass two dimension parameters to create a 4×4 matrix:

matrix(4, 4, [[1, 2], ]) In this case, the dimension of the matrix is given by the dimension parameters. As before, missing entries in an inner list correspond to zero, and in addition missing rows are treated as zero rows.

If the dimension m×n of the matrix is stated explicitly, the entries may also be specified by a plain list with mn elements. The matrix is filled with these elements row by row:

matrix(2, 3, [1, 2, 3, 4, 5, 6]) matrix(3, 2, [1, 2, 3, 4, 5, 6]) Example 8

A one- or two-dimensional array of arithmetical expressions, such as:

a := array(1..3, 2..4,
[[1, 1/3, 0], [-2, 3/5, 1/2], [-3/2, 0, -1]]
) can be converted into a matrix as follows:

A := matrix(a) Arrays serve, for example, as an efficient structured data type for programming. However, arrays do not have any algebraic meaning, and no mathematical operations are defined for them. If you convert an array into a matrix, you can use the full functionality defined for matrices as described above. For example, let us compute the matrix 2 A - A2 and the Frobenius norm of A:

2*A - A^2, norm(A, Frobenius) Note that an array may contain uninitialized entries:

b := array(1..4): b := 2: b := 0: b matrix cannot handle arrays that have uninitialized entries, and responds with an error message:

matrix(b)
Error: Unable to define a matrix over 'Dom::ExpressionField()'. [(Dom::Matrix(Dom::ExpressionField()))::new]

We initialize the remaining entries of the array b and convert it into a matrix, or more precisely, into a column vector:

b := 0: b := -1: matrix(b) delete a, A, b:

Example 9

We show how to create a matrix whose components are defined by a function of the row and the column index. The entry in the i-th row and the j-th column of a Hilbert matrix (see also linalg::hilbert) is . Thus the following command creates a 2×2 Hilbert matrix:

matrix(2, 2, (i, j) -> 1/(i + j - 1)) The following two calls produce different results. In the first call, x is regarded as an unknown function, while it is a constant in the second call:

delete x:
matrix(2, 2, x), matrix(2, 2, (i, j) -> x) Example 10

Diagonal matrices can be created by passing the option Diagonal and a list of diagonal entries:

matrix(3, 4, [1, 2, 3], Diagonal) One can generate the 3×3 identity matrix as follows:

matrix::identity(3) Here are alternative ways to create this matrix:

matrix(3, 3, [1 \$ 3], Diagonal) Equivalently, you can use a function of one argument:

matrix(3, 3, i -> 1, Diagonal) Since the integer 1 also represents a constant function, the following shorter call creates the same matrix:

matrix(3, 3, 1, Diagonal) To demonstrate the use of tables for creating (sparse) matrices we can also create the identity matrix above by the lines:

t := table(): t[1, 1] := 1: t[2, 2] := 1: t[3, 3] := 1:
matrix(3, 3, t) delete t:

Example 11

Banded Toeplitz matrices can be created with the option Banded. The following command creates a tri-diagonal matrix with constant bands:

matrix(4, 4, [-1, 2, -1], Banded) Example 12

Matrices can also be created by using a table:

t := table():
t[1, 2] := 12:
t[3, 1] := 31:
t[3, 2] := 32:
t The missing table entries correspond to empty matrix entries:

A := matrix(4, 6, t) By using tables, one can easily create large (sparse) matrices without being forced to define all zero entries of the matrix. Note that this is a great advantage over using arrays where every component has to be initialized before.

delete t, A:

Example 13

The method "doprint" of Dom::Matrix() prints only the non-zero components of a sparse matrix:

A := matrix(4, 6):
A[1, 2]:= 12: A[3, 1]:= 31: A[3, 2]:= 32:
print(A::dom::doprint(A)): delete A:

Parameters

 Array A one- or two-dimensional array or hfarray List ListOfRows A nested list of rows, each row being a list of arithmetical expressions Matrix A matrix, i.e., an object of a data type of category Cat::Matrix Table A table of matrix components m The number of rows: a positive integer n The number of columns: a positive integer f A function or a functional expression of two arguments g A function or a functional expression of one argument i1, i2, … Row indices: integers between 1 and m j1, j2, … Column indices: integers between 1 and m value1, value2, … Matrix entries: arithmetical expressions

Options

 Diagonal Create a diagonal matrix With this option, diagonal matrices can be created with diagonal elements taken from a list, or computed by a function or a functional expression. matrix(m, n, List, Diagonal) creates the m×n diagonal matrix whose diagonal elements are the entries of List. Cf. Example 10. List must have no more than min(m, n) entries. If it has fewer elements, the remaining diagonal elements are regarded as zero. matrix(m, n, g, Diagonal) returns the sparse matrix whose i-th diagonal element is g(i, i), where the index i runs from 1 to min(m, n). Cf. Example 10. Banded Create a banded Toeplitz matrix A banded matrix has zero entries outside the main diagonal and some of the adjacent sub- and superdiagonals. matrix(m, n, List, Banded) creates an m×n banded Toeplitz matrix with the elements of List as entries. The number of entries of List must be odd, say 2 h + 1, where h must not exceed n. The bandwidth of the resulting matrix is at most h. All elements of the main diagonal of the created matrix are initialized with the middle element of List. All elements of the i-th subdiagonal are initialized with the (h + 1 - i)-th element of List. All elements of the i-th superdiagonal are initialized with the (h + 1 + i)-th element of List. All entries on the remaining sub- and superdiagonals are regarded as zero. Cf. Example 11.

Return Values

Matrix of the domain type Dom::Matrix().