Matlab Matrix Operations

Matrix Operations

• In this note, “matrices” means 2-D numeric arrays, including scalars and vectors.

MATLAB has two flavors of Matlab Arithmetic Operations: matrix operations and Matlab Array Operations.

Matrix operations are the matrix manipulations of Linear Algebra, as opposed to the element-by-element computations of Matlab Array Operations.

Operators

The basic unary and binary numeric matrix operations all have an operator form:

Operator	Equivalent Function	Description	Arguments
+	plus	addition	binary, compatible arrays
-	minus	subtraction	binary, compatible arrays
*	mtimes	multiplication	binary, matrices
/	mrdivide	right division	binary, matrices
\	mldivide	left division	binary, matrices
^	mpower	powers	binary, matrices
'	ctranspose	complex conjugate transpose	unary, matrices
.'	transpose	non-conjugate transpose	unary, matrices

Notes

• In the table above, “compatible arrays” means the operands may be N-D arrays as long as they satisfy Matlab Compatible Array Sizes
• “matrices” refers to ordinary matrix arithmetic and scalar arithmetic
  • For example, A * B requires the column count of A to equal the row count of B
  • Or c * B, A * c where c is a scalar
• Notice that every operation requiring matrix operands maps to a function name beginning with m, indicating “matrix operation”
• Addition + and subtraction - coincide with their array-operation counterparts
• Multiplication *, when one operand is a scalar, behaves like .*
• Right division B / A is ${\mathbf{BA}}^{-1}$
  • Equivalent to B * inv(A) using the inverse function inv
  • When A is a scalar, this reduces to ./
  • In particular, A may be non-square; in that case the operation is equivalent to using the pseudo-inverse pinv: B * pinv(A)
• Left division A \ B is ${\mathbf{A}}^{-1}B$
  • Equivalent to inv(A) * B
  • When A is a scalar, this reduces to .\
  • In particular, A may be non-square; in that case the operation is equivalent to pinv(A) * B
• Power requires at least one operand to be a scalar (denoted below by lowercase letters)
  • a ^ B is $e^{(\ln a)\mathbf{B}}$
    • Equivalent to the matrix exponential function expm: expm(log(a) * B)
    • ++Caution++
      • The power operator computes $\mathbf{V}{a}^{\mathbf{D}}{\mathbf{V}}^{-1}$, where $\mathbf{V}$ and $\mathbf{D}$ are the matrices of eigenvectors (as columns) and eigenvalues (on the diagonal) of $\mathbf{B}$, so that $\mathbf{B}=\mathbf{V}\mathbf{D}{\mathbf{V}}^{-1}$
      • This algorithm requires $\mathbf{B}$‘s eigenvectors to be linearly independent so that $\mathbf{V}$ is invertible
      • The function expm instead uses ${\sum }_{k=0}^{\infty }\frac{1}{k!}{\mathbf{B}}^k$, which has no such requirement on $\mathbf{B}$
      • Therefore ^ may give a result different from expm; in particular, ^ may raise an error
  • A ^ a is ${\lim }_{\mathbb{Q}\ni q\to a}{\mathbf{A}}^q$
  • a ^ b is equivalent to a .^ b
• The transpose .' is the only matrix operator that contains a dot . yet is not an array operation