Mathematics

Vector Products and 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.

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 a_ij and a_ji. 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.

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. The unconjugated complex transpose is denoted by z.', in analogy with the other array operations. 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