Examples Examples demonstrating the usage of the Math.Matrices functions
	toString Convert a matrix into its string representation
	isEqual Compare whether two Real matrices are identical
	solve Solve real system of linear equations A*x=b with a b vector (Gaussian elimination with partial pivoting)
	solve2 Solve real system of linear equations A*X=B with a B matrix (Gaussian elimination with partial pivoting)
	leastSquares Solve linear equation A*x = b (exactly if possible, or otherwise in a least square sense; A may be non-square and may be rank deficient)
	leastSquares2 Solve linear equation A*X = B (exactly if possible, or otherwise in a least square sense; A may be non-square and may be rank deficient)
	equalityLeastSquares Solve a linear equality constrained least squares problem
	LU LU decomposition of square or rectangular matrix
	LU_solve Solve real system of linear equations P*L*U*x=b with a b vector and an LU decomposition (from LU(..))
	LU_solve2 Solve real system of linear equations P*L*U*X=B with a B matrix and an LU decomposition (from LU(..))
	eigenValues Return eigenvalues and eigenvectors for a real, nonsymmetric matrix in a Real representation
	eigenValueMatrix Return real valued block diagonal matrix J of eigenvalues of matrix A (A=V*J*Vinv)
	singularValues Return singular values and left and right singular vectors
	QR Return the QR decomposition of a square matrix with optional column pivoting (A(:,p) = Q*R)
	hessenberg Return upper Hessenberg form of a matrix
	realSchur Return the real Schur form (rsf) S of a square matrix A, A=QZ*S*QZ'
	cholesky Return the Cholesky factorization of a symmetric positive definite matrix
	balance Return a balanced form of matrix A to improve the condition of A
	balanceABC Return a balanced form of a system [A,B;C,0] to improve its condition by a state transformation
	trace Return the trace of matrix A, i.e., the sum of the diagonal elements
	det Return determinant of a matrix (computed by LU decomposition; try to avoid det(..))
	inv Return inverse of a matrix (try to avoid inv(..))
	rank Return rank of a rectangular matrix (computed with singular values)
	conditionNumber Return the condition number norm(A)*norm(inv(A)) of a matrix A
	rcond Return the reciprocal condition number of a matrix
	norm Return the p-norm of a matrix
	frobeniusNorm Return the Frobenius norm of a matrix
	nullSpace Return the orthonormal nullspace of a matrix
	exp Return the exponential of a matrix by adaptive Taylor series expansion with scaling and balancing
	integralExp Return the exponential and the integral of the exponential of a matrix
	integralExpT Return the exponential, the integral of the exponential, and time-weighted integral of the exponential of a matrix
	continuousLyapunov Return solution X of the continuous-time Lyapunov equation X*A + A'*X = C
	continuousSylvester Return solution X of the continuous-time Sylvester equation A*X + X*B = C
	continuousRiccati Return solution X of the continuous-time algebraic Riccati equation A'*X + X*A - X*B*inv(R)*B'*X + Q = 0 (care)
	discreteLyapunov Return solution X of the discrete-time Lyapunov equation A'*X*A + sgn*X = C
	discreteSylvester Return solution of the discrete-time Sylvester equation A*X*B + sgn*X = C
	discreteRiccati Return solution of discrete-time algebraic Riccati equation A'*X*A - X - A'*X*B*inv(R + B'*X*B)*B'*X*A + Q = 0 (dare)
	sort Sort the rows or columns of a matrix in ascending or descending order
	flipLeftRight Flip the columns of a matrix in left/right direction
	flipUpDown Flip the rows of a matrix in up/down direction
	LAPACK Interface to LAPACK library (should usually not directly be used but only indirectly via Modelica.Math.Matrices)
	Utilities Utility functions that should not be directly utilized by the user

This information is part of the Modelica Standard Library maintained by the Modelica Association.

Library content

This library provides functions operating on matrices. Below, the functions are ordered according to categories and a typical call of the respective function is shown. Most functions are solely an interface to the external LAPACK library.

Note: A' is a short hand notation of transpose(A):

Basic Information

• toString(A) - returns the string representation of matrix A.
• isEqual(M1, M2) - returns true if matrices M1 and M2 have the same size and the same elements.

Linear Equations

• solve(A,b) - returns solution x of the linear equation A*x=b (where b is a vector, and A is a square matrix that must be regular).
• solve2(A,B) - returns solution X of the linear equation A*X=B (where B is a matrix, and A is a square matrix that must be regular)
• leastSquares(A,b) - returns solution x of the linear equation A*x=b in a least squares sense (where b is a vector and A may be non-square and may be rank deficient)
• leastSquares2(A,B) - returns solution X of the linear equation A*X=B in a least squares sense (where B is a matrix and A may be non-square and may be rank deficient)
• equalityLeastSquares(A,a,B,b) - returns solution x of a linear equality constrained least squares problem: min|A*x-a|^2 subject to B*x=b
• (LU,p,info) = LU(A) - returns the LU decomposition with row pivoting of a rectangular matrix A.
• LU_solve(LU,p,b) - returns solution x of the linear equation L*U*x[p]=b with a b vector and an LU decomposition from "LU(..)".
• LU_solve2(LU,p,B) - returns solution X of the linear equation L*U*X[p,:]=B with a B matrix and an LU decomposition from "LU(..)".

Matrix Factorizations

• (eval,evec) = eigenValues(A) - returns eigen values "eval" and eigen vectors "evec" for a real, nonsymmetric matrix A in a Real representation.
• eigenValueMatrix(eval) - returns real valued block diagonal matrix of the eigenvalues "eval" of matrix A.
• (sigma,U,VT) = singularValues(A) - returns singular values "sigma" and left and right singular vectors U and VT of a rectangular matrix A.
• (Q,R,p) = QR(A) - returns the QR decomposition with column pivoting of a rectangular matrix A such that Q*R = A[:,p].
• (H,U) = hessenberg(A) - returns the upper Hessenberg form H and the orthogonal transformation matrix U of a square matrix A such that H = U'*A*U.
• realSchur(A) - returns the real Schur form of a square matrix A.
• cholesky(A) - returns the cholesky factor H of a real symmetric positive definite matrix A so that A = H'*H.
• (D,Aimproved) = balance(A) - returns an improved form Aimproved of a square matrix A that has a smaller condition as A, with Aimproved = inv(diagonal(D))*A*diagonal(D).

Matrix Properties

• trace(A) - returns the trace of square matrix A, i.e., the sum of the diagonal elements.
• det(A) - returns the determinant of square matrix A (using LU decomposition; try to avoid det(..))
• inv(A) - returns the inverse of square matrix A (try to avoid, use instead "solve2(..) with B=identity(..))
• rank(A) - returns the rank of square matrix A (computed with singular value decomposition)
• conditionNumber(A) - returns the condition number norm(A)*norm(inv(A)) of a square matrix A in the range 1..∞.
• rcond(A) - returns the reciprocal condition number 1/conditionNumber(A) of a square matrix A in the range 0..1.
• norm(A) - returns the 1-, 2-, or infinity-norm of matrix A.
• frobeniusNorm(A) - returns the Frobenius norm of matrix A.
• nullSpace(A) - returns the null space of matrix A.

Matrix Exponentials

• exp(A) - returns the exponential e^A of a matrix A by adaptive Taylor series expansion with scaling and balancing
• (phi, gamma) = integralExp(A,B) - returns the exponential phi=e^A and the integral gamma=integral(exp(A*t)*dt)*B as needed for a discretized system with zero order hold.
• (phi, gamma, gamma1) = integralExpT(A,B) - returns the exponential phi=e^A, the integral gamma=integral(exp(A*t)*dt)*B, and the time-weighted integral gamma1 = integral((T-t)*exp(A*t)*dt)*B as needed for a discretized system with first order hold.

Matrix Equations

• continuousLyapunov(A,C) - returns solution X of the continuous-time Lyapunov equation X*A + A'*X = C
• continuousSylvester(A,B,C) - returns solution X of the continuous-time Sylvester equation A*X + X*B = C
• continuousRiccati(A,B,R,Q) - returns solution X of the continuous-time algebraic Riccati equation A'*X + X*A - X*B*inv(R)*B'*X + Q = 0
• discreteLyapunov(A,C) - returns solution X of the discrete-time Lyapunov equation A'*X*A + sgn*X = C
• discreteSylvester(A,B,C) - returns solution X of the discrete-time Sylvester equation A*X*B + sgn*X = C
• discreteRiccati(A,B,R,Q) - returns solution X of the discrete-time algebraic Riccati equation A'*X*A - X - A'*X*B*inv(R + B'*X*B)*B'*X*A + Q = 0

Matrix Manipulation

• sort(M) - returns the sorted rows or columns of matrix M in ascending or descending order.
• flipLeftRight(M) - returns matrix M so that the columns of M are flipped in left/right direction.
• flipUpDown(M) - returns matrix M so that the rows of M are flipped in up/down direction.

See also

Vectors