Syntax

A+B 
A-B
A*B     A.*B
A/B     A./B
A\B     A.\B
A^B     A.^B
A'      A.'

Description

+	Addition or unary plus. A+B adds A and B. A and B must have the same size, unless one is a scalar. A scalar can be added to a matrix of any size.
-	Subtraction or unary minus. A-B subtracts B from A. A and B must have the same size, unless one is a scalar. A scalar can be subtracted from a matrix of any size.
*	Matrix multiplication. C = A*B is the linear algebraic product of the matrices A and B. More precisely, For nonscalar A and B, the number of columns of A must equal the number of rows of B. A scalar can multiply a matrix of any size.
.*	Array multiplication. A.*B is the element-by-element product of the arrays A and B. A and B must have the same size, unless one of them is a scalar.
/	Slash or matrix right division. B/A is roughly the same as B*inv(A). More precisely, B/A = (A'\B')'. See \.
./	Array right division. A./B is the matrix with elements A(i,j)/B(i,j). A and B must have the same size, unless one of them is a scalar.
\	Backslash or matrix left division. If A is a square matrix, A\B is roughly the same as inv(A)*B, except it is computed in a different way. If A is an n-by-n matrix and B is a column vector with n components, or a matrix with several such columns, then X = A\B is the solution to the equation AX = B computed by Gaussian elimination (see "Algorithm" for details). A warning message prints if A is badly scaled or nearly singular.
	If A is an m-by-n matrix with m ~= n and B is a column vector with m components, or a matrix with several such columns, then X = A\B is the solution in the least squares sense to the under- or overdetermined system of equations AX = B. The effective rank, k, of A, is determined from the QR decomposition with pivoting (see "Algorithm" for details). A solution X is computed which has at most k nonzero components per column. If k < n, this is usually not the same solution as pinv(A)*B, which is the least squares solution with the smallest norm, ||X||.
.\	Array left division. A.\B is the matrix with elements B(i,j)/A(i,j). A and B must have the same size, unless one of them is a scalar.
^	Matrix power. X^p is X to the power p, if p is a scalar. If p is an integer, the power is computed by repeated multiplication. If the integer is negative, X is inverted first. For other values of p, the calculation involves eigenvalues and eigenvectors, such that if [V,D] = eig(X), then X^p = V*D.^p/V.
	If x is a scalar and P is a matrix, x^P is x raised to the matrix power P using eigenvalues and eigenvectors. X^P, where X and P are both matrices, is an error.
.^	Array power. A.^B is the matrix with elements A(i,j) to the B(i,j) power. A and B must have the same size, unless one of them is a scalar.
'	Matrix transpose. A' is the linear algebraic transpose of A. For complex matrices, this is the complex conjugate transpose.
.'	Array transpose. A.' is the array transpose of A. For complex matrices, this does not involve conjugation.

Remarks

Examples

Algorithm

• If A is a triangular matrix, or a permutation of a triangular matrix, then X can be computed quickly by a permuted backsubstitution algorithm. The check for triangularity is done for full matrices by testing for zero elements and for sparse matrices by accessing the sparse data structure. Most nontriangular matrices are detected almost immediately, so this check requires a negligible amount of time.
• If A is symmetric, or Hermitian, and has positive diagonal elements, then a Cholesky factorization is attempted (see chol). If A is sparse, a symmetric minimum degree preordering is applied (see symmmd and spparms). If A is found to be positive definite, the Cholesky factorization attempt is successful and requires less than half the time of a general factorization. Nonpositive definite matrices are usually detected almost immediately, so this check also requires little time. If successful, the Cholesky factorization is

  A = R'*R
  

  where R is upper triangular. The solution X is computed by solving two triangular systems,

  X = R\(R'\B)
  

• If A is square, but not a permutation of a triangular matrix, or is not Hermitian with positive elements, or the Cholesky factorization fails, then a general triangular factorization is computed by Gaussian elimination with partial pivoting (see lu). If A is sparse, a nonsymmetric minimum degree preordering is applied (see colmmd and spparms). This results in

  A = L*U
  

  where L is a permutation of a lower triangular matrix and U is an upper triangular matrix. Then X is computed by solving two permuted triangular systems.

  X = U\(L\B)
  

• If A is not square and is full, then Householder reflections are used to compute an orthogonal-triangular factorization.

  A*P = Q*R
  

  where P is a permutation, Q is orthogonal and R is upper triangular (see qr). The least squares solution X is computed with

  X = P*(R\(Q'*B)
  

• If A is not square and is sparse, then the augmented matrix is formed by:

  S = [c*I A; A' 0]
  

  The default for the residual scaling factor is c = max(max(abs(A)))/1000 (see spparms). The least squares solution X and the residual R = B-A*X are computed by

  S * [R/c; X] = [B; 0]
  

  with minimum degree preordering and sparse Gaussian elimination with numerical pivoting.

Diagnostics

Matrix is singular to working precision.

Divide by zero.

Warning: Matrix is close to singular or badly scaled.
    Results may be inaccurate.  RCOND = xxx

Warning: Rank deficient, rank = xxx tol = xxx

See Also

inv         Matrix inverse

lu          LU matrix factorization

orth        Range space of a matrix

qr          Orthogonal-triangular decomposition

rref        Reduced row echelon form

References