CHAPTER 4

Symbolic Matrix Algebra

4-1. Vectors and Matrices

In the preceding chapter’s coverage of vector and matrix variables, we saw how to define vectors and matrices in MATLAB. At the same time, we defined simple operations with vector and matrix variables. This chapter will expand the concepts of matrix algebra, introducing commands that allow you to work with matrices.

Consider the matrix:

You can enter this in MATLAB in any of the following ways:

A=[a11,a12,...,a1n ; a21,a22,...,a2n ; ... ; am1,am2,...,amn]

A=[a11 a12 ... a1n ; a21 a22 ... a2n ; ... ; am1 am2 ... amn]

A=maple('array([[a11,..,a1n],[a21,..,a2n],..,[am1,..,amn]])')

A=maple('matrix(m,n,[a11,..,a1n,a21,..,a2n,..,am1,..,amn])')

A=maple('matrix([[a11,..,a1n],[a21,..,a2n],..,[am1,..,amn]])')

At the same time, consider the vector

V =(v1,v2,...,vn)

This is a particular case of a matrix, consisting of a single row (i.e. it is a matrix of dimension 1×n). One can define it in any of the following ways in MATLAB:

V = [v1, v2,..., vn]

V = [v1 v2... vn]

V = maple('vector([v1, v2,..., vn])')

V = maple('vector(n,[v1, v2,..., vn])')

V=maple('array([v1, v2, ..., vn])')

4-2. Operations with Symbolic Matrices

MATLAB supports most matrix algebra operations (sum, difference, product, scalar multiplication). Some operations can always be applied while others depend on meeting certain dimensionality criteria.

The following MATLAB commands allow operations with matrices.

• A + B gives the sum of matrices A and B.
• A - B gives the difference between the matrices A and B (A minus B).
• c * M gives the product of the scalar c and the matrix M.
• A * B gives the product of the matrices A and B (A B).
• A ^ p gives the matrix A raised to the power of the scalar p.
• p ^ A gives p raised to the matrix A.
• expm(A) gives e^A calculated via eigenvalues.
• expm1(A) gives e^A calculated via Pade approximants.
• expm2(A) gives e^A calculated via Taylor series.
• expm3(A) gives e^A calculated via the condition number of the matrix of eigenvectors.
• logm(A) gives the Napierian logarithm of matrix A.
• sqrtm(A) gives the square root of the square matrix A.
• funm(A,'function') applies the function to the square matrix A.
• transpose(A) or  A' gives the transpose of the matrix A.
• inv(A) gives the inverse of the square matrix A (i.e. the matrix A^{- 1}).
• det(A) gives the determinant of the square matrix A.
• rank(A) gives the rank of the matrix A.
• trace(A) gives the sum of the elements of the diagonal of A.
• Svd(A) gives the vector V of singular values of A. The singular values of A are the square roots of the eigenvalues of the symmetric matrix A' A.
• [U,S,V] = Svd(A) gives the diagonal matrix S of singular values of A (ordered in decreasing magnitude), and the matrices U and V such that = U * S * V'.
• cond(A) gives the condition number of the matrix A (the ratio between the largest and the smallest singular values of A).
• rcond(A) gives the reciprocal of the condition number of the matrix A .
• norm(A) gives the norm of A (the greatest singular value of the matrix A).
• norm(A,1) gives the 1-norm of A (the maximum column sum of A, where the column sum is the sum of the absolute values of the entries in a column).
• norm(A,inf) gives the infinity norm of A (the maximum row sum of A, where the row sum is the sum of the absolute values of the entries in a row).
• norm(A,'fro') gives the Frobenius norm of A, defined by sqrt(sum(diag(A'A))).
• Z = null(A) gives an orthonormal basis of the kernel of A (so that Z'Z = I). The number of columns of Z is the nullity of A.
• Q = orth(A) gives an orthonormal basis of the range of A (so that Q'Q = I). The columns of Q generate the same space as the columns of A, and the number of columns in Q is the rank of A.
• subspace(A,B) gives the angle between the subspaces specified by the columns of A and B.
• rref(A) produces the row reduced echelon form of A. The number of non-zero rows of rref(A) is the rank of the matrix A.

4-3. Other Symbolic Matrix Operations

MATLAB also provides the following commands that allow operations with symbolic matrices. (The maple command is required to define symbolic matrices):

• A = sym('[f1;f2;...;fm]') defines the symbolic m × n matrix with rows f1 to fm, where fi = ai1, ai2,..., ain.
• symadd(A,B) gives the sum of matrices A and B (A plus B).
• symsub(A,B) gives the difference of the matrices A and B (A minus B).
• symmul(A,B) gives the product of matrices A and B (A B).
• sympow(A,p) gives A raised to the power of the scalar p.
• transpose(A) gives the transpose of the matrix A (A').
• inv(A) gives the inverse of square matrix A (A^-1).
• det(A) gives the determinant of the square matrix A.
• rank(A) gives the rank of the matrix A.
• Svd(A) or singvals(A) gives the vector of singular values of A. The singular values of A are the square roots of the eigenvalues of the symmetric matrix A ' A.
• [U,S,V] = singvals(A) or [U,S,V] = Svd(A) returns the orthogonal matrices U and V and the diagonal matrix S with singular values of A on the diagonal, such that A= USV'.
• symop(A,'operation1',B,'operation2',C,...) performs the specified operations between the given symbolic matrices and in the order given. This command allows you to mix all kinds of operations between symbolic matrices.
• maple('evalm(expr(A,B,C,..))') evaluates the expression in the matrices A, B, C,... This expression has to be formed by the basic operators addition (+), subtraction (-), product (& *) and power (^). Within evalm the zero matrix is denoted by 0, the identity matrix is denoted by & * () and the inverse matrix is denoted by A ^(-1). In addition, A ^ 0 is always 1.
• maple('matadd(A,B)') adds the matrices or vectors A and B (A+B).
• maple('matadd(A,B,k,r)') calculates k * A + r * B.
• maple(scalarmul(A,k)) calculates k * A (scalar multiple).
• maple('multiply(A,B,C,...)')or maple('A&*B&*C&*...') computes the product of the given matrices in the order specified.
• maple(exponential(A,t)) calculates e^At via Taylor series. It can be stated as e ^At = I + At + 1/2!A²t²+ . . . .
• maple('exponential(A)') calculates e ^Ax where x is the first variable found within the matrix A.
• maple('transpose(A)') gives the transpose of the matrix or vector A (A' ).
• maple('htranspose(A)') gives the hermitian transpose of the matrix or vector A. Its (i, j)th element is defined as the conjugate of the (j, i)th element of A.
• maple('inverse(A)') or maple(evalm(A^(-1))) finds the inverse of the square matrix A  (i.e. A^{- 1}).
• maple('adjoint(A)') or maple('adj(A)') finds the adjoint of the square matrix A (adjoint(A) = inverse(A) * det(A)).
• maple('minor(A,i,j)') returns the determinant of the matrix obtained by deleting the i-th row and j-th column of the matrix A.
• maple('det(A)') returns the determinant of the square matrix A.
• maple(det(A,sparse)) returns the determinant of a sparse square matrix A, calculated by an efficient method of minor expansion.
• maple('Det(A)') returns the inert determinant of the square matrix A.
• maple(Det(A) mod n) returns the determinant of A modulo n.
• maple('permanent(A)') calculates the permanent of the matrix A (similar to the calculation of the determinant of A but such that there are no alternating signs in the terms of the sum).
• maple('rank(A)') returns the rank of the matrix A.
• maple('trace(A)') returns the sum of the elements of the diagonal of A.
• maple('Svd(A)') or maple('(singularvals(A)') gives the vector of singular values of A. The singular values of A are the square roots of the eigenvalues of the symmetric matrix A' A.
• norm(A) or norm(A,2) returns the standard norm of A defined as the maximum of the singular values of A.
• norm(A,inf) or maple('norm(A)') gives the infinity norm of A defined as the maximum of the row sums of A (where a row sum is the sum of the absolute values of the entries of the row).
• norm(A,1) gives 1-norm of A defined by the maximum of the column sums of A (where a column sum is the sum of the absolute values of the entries of the column).
• norm(A,fro) gives the Frobenius norm of A defined as sqrt(sum(diag(A'*A))).
• maple('norm(A,option)') gives the norm of A according to the given option. Possible values for option are 1, 2, infinity and frobenius. The 1-norm is the maximum of the column sums of A. The 2-norm is the square root of the largest eigenvalue of AA'. The infinity norm  is the maximum of the row sums of A. The Frobenius norm is the square root of the sum of the squares of the elements of A.
• normest(A) estimates the 2-norm of A.
• cond(A) or maple('cond(A)') gives the condition number of the matrix A (the product of the infinity norm of A and the infinity norm of A^-1, or the ratio between the largest and the smallest singular values of A).
• cond(A,P) gives norm(X,P) * norm(inv(X),P) where P is set to 1, 2, infinity or fro according to the type of norm one wants to use.
• maple('cond(A,option)') returns the condition number of A according to the given option. Possible values for option are 1, 2, infinity and frobenius. The options determine the norm used in the calculation of the condition number.
• [C,V] = condest(A) gives C and V such that the condition norm(A*V,1) = norm(A, 1) * norm(V,1)/c is met.
• maple('orthog(A)') determines whether A is an orthogonal matrix (i.e. if A^{- 1} = A' ).
• maple('diag(A1,A2,...,An)') or maple ('BlockDiagonal(A1,A2,...,An)') builds the diagonal matrix whose diagonal elements are the subarrays (or elements) A1, A2,..., An.
• maple('blockmatrix(m,n,B11,...,B1m,...,Bn1,...,Bnm)') constructs an m × n matrix with the blocks given, taken in consecutive order.
• maple('diag(V,n)') creates a square matrix of dimension n with diagonal elements given by the vector V.
• maple('submatrix(A,i..k,j...h)') extracts from A the subarray formed by rows i to k, and by columns j to h.
• maple('subvector(A,i,j1..j2)') extracts from the matrix A the subvector determined by the ith  row between columns j1 and j2, both inclusive.
• maple('subvector(A,i1..i2,j)') extracts from the matrix A the subvector determined by the jth column between rows i1 and i2, both inclusive.
• maple('row(A,i)') extracts row i from A.
• maple('row(A,i..k)') extracts from A the rows from i to k.
• maple('column(A,j)') extracts column j from A.
• maple('column(A,j..h)') extracts from  A the columns from j to h.
• maple('addcol(A,j1,j2,expr)')creates a new array, replacing column j2 of A by expr * colummnj1 + columnj2.
• maple('addrow(A,i1,i2,expr)') creates a new array, replacing row i2 of A by expr * rowi1 + rowi2.
• maple('mulcol(A,j,expr)') creates a new matrix by multiplying column j of A by the expression expr.
• maple('mulrow(A,i,expr)') creates a new matrix by multiplying row i of A by the expression expr.
• maple('swapcol(A,j1,j2)') exchanges columns j1 and j2 of A.
• maple('swaprow(A,i1,i2)') exchanges rows i1 and i2 of A.
• maple('stack(A,B)') creates a new array by placing A over B (A and B have the same number of columns).
• maple('augment(A,B)') or maple('concat(A,B)') creates a new array by placing the array A to the left of B (A and B have the same number of rows).
• maple('extend(A,m,n)') creates a new array by adding m rows and n columns to A matrix, leaving unassigned new elements.
• maple('extend(A,m,n,expr)') creates a new array by adding m rows and n columns to the matrix A, filling the new elements with expr.
• maple('copyinto(A,B,i,j)') updates B by copying elements of A into B starting at element B(i,j).
• maple('pivot(A,i,j)') pivots the matrix A on its element A_ij .
• maple('pivot(A,i,j,ia...ib)') pivots the matrix A on its element A_ij  but only modifies the rows between ia and ib.
• maple('(rowdim(A)') gives the number of rows in A.
• maple('coldim(A)') gives the number of columns in A.
• maple('(vectdim(V)') gives the dimension of the vector V.
• maple('delrows(A,i..k)') deletes rows i to k from A.
• maple('delcols(A,j...h)') deletes columns i to k from A .
• maple('equal(A,B)') determines whether the matrices or vectors A and B are equal.
• maple('issimilar(A,B)') determines whether the matrices or vectors A and B are similar. A and B are similar if there exists an M such that A = evalm(inverse(M) & * B & * M).
• maple('issimilar(A,B,name)') assigns name to the matrix such that A = evalm(inverse(name) & * B & * name).
• maple('iszero(A)') determines whether the matrix A is the zero matrix.
• null(A) or maple('kernel(A)') or maple('nullspace(A)') gives a set of vectors that span the kernel of the linear transformation defined by the matrix A.
• expm(A) finds e^A according to Padé’s algorithm.
• expm1(A) finds e^A according to Golub’s algorithm.
• expm2(A) finds e^A via Taylor series.
• expm3(A) finds e^A via eigenvalues and eigenvectors.
• diag(V,k) builds a diagonal square matrix of order n + |k| with the n elements of the vector V in the kth diagonal. If k = 0, the diagonal is the main diagonal, if k > 0, the diagonal is k places above the main diagonal, and if k < 0, the diagonal is k places below the main diagonal. We have diag(V,0) = diag(V).
• triu(A,k) constructs an upper triangular matrix with elements of A that are above the kth diagonal. If k = 0, the diagonal is the main diagonal, if k > 0, the diagonal is k places above the main diagonal, and if k < 0, the diagonal is k places below the main diagonal. We have triu(A,0) = triu(A).
• tril(A,k) builds a lower triangular matrix with elements of A that are below the kth diagonal. If k = 0, the diagonal is the main diagonal, if k > 0, the diagonal is k places above the main diagonal, and if k < 0, the diagonal is k places below the main diagonal.. We have, tril(A,0) = tril(A).
• rref(A) or rrefmovie(A) produces the row reduced echelon form of the matrix A.
• colspace(A) gives a basis for the vector space generated by the columns of the matrix A.
• Q = orth(A) gives an orthonormal basis for the range of A, that is, Q'Q = I and the columns of Q generate the same space as the columns of A, where the number of columns in Q is equal to the rank of A .
• maple('randmatrix(m,n)') generates a random matrix of order (m × n). The elements are by default between –99 and 99, but this range can be changed via the command rand(a..b) which yields a random number between a and b. The command readlib(randomize): randomize(n) is used to set the generating seed for the value randomize(n) (by default a seed generated by the system clock is used).
• maple('randmatrix(m,n,option)') generates a random matrix of order (m × n) according to the specified option. The options can be symmetric, antisymmetric, diagonal, unimodular and sparse, depending on whether the random matrix to be generated is symmetric, antisymmetric, diagonal, unimodular, or sparse, respectively.
• maple('randvector(n)') generates a random vector of length n.
• maple('(entermatrix(A)') enables an interface to input values of a matrix, separating elements by commas. You first need to specify the dimension of the array with the command A = matrix(m,n).
• maple('array(1.. m,1... n,[(1,1)=a11,...,(m,n)=amn],option)') specifies the array of dimension (m × n) according to the option specified. The options can be symmetric, antisymmetric, diagonal, identity and sparse, depending on the type of array you want to define.
• maple('matrix(m,n,f)') defines a matrix of dimension m × n whose elements are those specified by the function f(i,j) i = 1... m, j = 1... n.
• maple('vector(n,f)') defines the vector of dimension n whose elements are those specified by the function f(i) i = 1... n.
• maple('array(identity,1..n,1..n) ') defines the  n × n identity matrix.

Some examples follow. First, let’s consider three alternative ways of defining the same symbolic matrix (let’s not forget the maple command, which is always needed to define symbolic matrices and vectors):

>> A = sym('[1,2,3;4,5,6;7,8,9]')

A =

[1, 2, 3]
[4, 5, 6]
[7, 8, 9]

>> A = sym(maple('array([[1,2,3],[4,5,6],[7,8,9]])'))

A =

[1, 2, 3]
[4, 5, 6]
[7, 8, 9]

>> A = sym(maple('matrix([[1,2,3],[4,5,6],[7,8,9]])'))

A =

[1, 2, 3]
[4, 5, 6]
[7, 8, 9]

>> A = sym(maple('matrix(3,3,[1,2,3,4,5,6,7,8,9])'))

A =

[1, 2, 3]
[4, 5, 6]
[7, 8, 9]

Next we define a symbolic matrix by using a function that defines its elements, in particular A_{i, j} = 1 / (i+j).

>> A = sym(maple('matrix(3,3,(i,j)->1/(i+j))'))

A =

[1/2, 1/3, 1/4]
[1/3, 1/4, 1/5]
[1/4, 1/5, 1/6]

Now let’s define the third-order identity matrix in two different ways:

>> A = sym(maple('array(1..3,1..3,identity)'))

A =

[1, 0, 0]
[0, 1, 0]
[0, 0, 1]

>> A = sym(maple('matrix(3,3,(i,j)->if i=j then 1 else 0 fi)'))

A =

[1, 0, 0]
[0, 1, 0]
[0, 0, 1]

Next we define sparse, symmetric and antisymmetric matrices:

>> sym(maple('array(1..3,1..3,[(1,1)=1,(1,2)=2,(1,3)=3,(2,2)=4,(2,3)=6,(3,3)=5], symmetric)'))

ans =

[1, 2, 3]
[2, 4, 6]
[3, 6, 5]

>> sym(maple('array(1..3,1..3,[(1,1)=1,(1,2)=2,(1,3)=3,(2,2)=4,(2,3)=6,(3,3)=5], sparse)'))

ans =

[1, 2, 3]
[0, 4, 6]
[0, 0, 5]

>> sym(maple('array(1..3,1..3,[(1,2)=2,(1,3)=3,(2,3)=4], antisymmetric)'))

ans =

[0,  2,  3]
[-2,  0,  4]
[-3, -4,  0]

>> sym(maple('array(1..5,1..7,[(2,3)=4,(5,5)=12], sparse)'))

ans =

[0, 0, 0, 0, 0, 0, 0]
[0, 0, 4, 0, 0, 0, 0]
[0, 0, 0, 0, 0, 0, 0]
[0, 0, 0, 0, 0, 0, 0]
[0, 0, 0, 0, 12, 0, 0]

Next we define in different ways the symbolic vector whose components are the first six integers.

>> pretty(sym('[1,2,3,4,5,6]'))

>> pretty(sym(maple('vector([1,2,3,4,5,6])')))

>> pretty(sym(maple('vector(6,[1,2,3,4,5,6])')))

>> pretty(sym(maple('array([1,2,3,4,5,6])')))

>> pretty(sym(maple('vector(6,i->i)')))

4-4. Eigenvalues and Eigenvectors: Diagonalization

MATLAB provides the following commands that allow you to work with eigenvalues and eigenvectors of a square matrix:

• eig(A) calculates the eigenvalues of the square matrix A.
• [V,D] = eig(A) calculates the diagonal matrix D of eigenvalues of A and a matrix V whose columns are the corresponding eigenvectors, such that A * V = V * D.
• eig(A,B) gives a vector that contains the generalized eigenvalues of the square matrices A and B. The generalized eigenvalues of A and B are the roots of the polynomial in λ, det (λ * B - A).
• [V,D] = eig(A,B) calculates the diagonal matrix D of generalized eigenvalues of  A and B, and an array V whose columns are the corresponding eigenvectors, such that A * V = B * V * D.
• [AA,BB,Q,Z,V] = qz(A,B) calculates the upper triangular matrices AA and BB and the matrices Q and Z such that Q * A * Z = AA and Q * B * Z = BB, and gives the matrix V of generalized eigenvectors of A and B. The generalized eigenvalues are the elements of the diagonals of AA and BB, such that A * V * diag(BB) = B * V * diag(AA).
• [T,B] = balance(A) returns a similarity transformation T and a matrix B such that B = TA * T has eigenvalues approximately equal to those of A. The matrix B is called the balanced matrix of the matrix A.
• balance(A) calculates the balanced matrix B of the matrix A. Its use is essentially to approximate the eigenvalues of A when they are difficult to estimate. We have eig(A) = eig(balance(A)).
• [V,D] = cdf2rdf(V,D) converts a complex diagonal form to a real block diagonal form. Each complex eigenvalue in the diagonal of the input D becomes a 2 x 2 subarray in the transformed matrix D.
• [U,T] = schur(A) gives a matrix T and a unitary matrix U such that A = U * T * U' and U'* U = eye(U). If A is complex, T is an upper triangular matrix with the eigenvalues of A on its diagonal. If A  is real, the matrix T has the eigenvalues of A on its diagonal, and complex eigenvalues will correspond to 2 x 2 diagonal blocks on the diagonal of T. The command schur(A) returns the matrix T only.
• [U,T] = rsf2csf(U,T) converts a real Schur form to a complex Schur form.
• [H,p] = hess(A) returns the unitary matrix P and Hessenberg matrix H such that A = P * H * P' and P'* P = eye(size(P)).
• hess(A) returns the Hessenberg matrix H of  A.
• poly(A) returns the characteristic polynomial of the matrix A.
• poly(V) returns a vector whose components are the coefficients of the polynomial whose roots are the elements of the vector V.
• vander(C) returns a Vandermonde matrix such that its j-th column is A(:,j) = C ^ (n-j).