Mathematics

Vector Productsand Transpose

A row vector and a column vector of the same length can be multiplied in either order. The result is either a scalar, the inner product, or a matrix, the outer product.

• u = [3; 1; 4];
  v = [2 0 -1];
  x = v*u
  
  x =
         2
  
  X = u*v
  
  X =
         6     0     -3
         2     0     -1
         8     0     -4
  

For real matrices, the transpose operation interchanges and . MATLAB uses the apostrophe (or single quote) to denote transpose. Our example matrix A is symmetric, so A' is equal to A. But B is not symmetric.

• B = magic(3);
  X = B'
  
  X =
         8     3     4
         1     5     9
         6     7     2
  

Transposition turns a row vector into a column vector.

• x = v'
  
  x =
         2
         0
        -1
  

If x and y are both real column vectors, the product x*y is not defined, but the two products

• x'*y
  

and

• y'*x
  

are the same scalar. This quantity is used so frequently, it has three different names: inner product, scalar product, or dot product.

For a complex vector or matrix, z, the quantity z' denotes the complex conjugate transpose, where the sign of the complex part of each element is reversed. The unconjugated complex transpose, where the complex part of each element retains its sign, is denoted by z.'. So if

• z = [1+2i 3+4i]
  

then z' is

• 1-2i
  3-4i
  

while z.' is

• 1+2i
  3+4i
  

For complex vectors, the two scalar products x'*y and y'*x are complex conjugates of each other and the scalar product x'*x of a complex vector with itself is real.

Addition and Subtraction

Matrix Multiplication