Mathematics - The MathWorks - (Version JPG) - Page n° 19 - PDF Catalogue | Technical Documentation

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Matrices in the MATLAB® Environment

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' not only transposes the vector or matrix, but also converts each complex element to its complex conjugate. That is, the sign of the imaginary part of each complex element is changed. So if

[1+2i 7-3i 3+4i; 6-2i 9i 4+7i]

z = z =

1.0000 6.0000

2.0000i 2.0000i

7.0000 -

3.0000i 9.0000i

3.0000 4.0000

4.0000i 7.0000i

then

z' ans

1.0000 7.0000 3.0000

2.0000i 3.0000i 4.0000i

6.0000

4.0000

2.0000i 9.0000i 7.0000i

The unconjugated complex transpose, where the complex part ofeach element retains its sign, is denoted by z.':

z. '

ans

1.0000 + 2.0000i 6.0000 - 2.0000i 7.0000 - 3.0000i 0 + 9.0000i

3.0000 + 4.0000i 4.0000 + 7.0000i

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.

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Mathematics - The MathWorks - #19

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