Tac-Tics said:Try it out for yourself. Take [tex]\mathbf{h} = [1, 0]^*[/tex].
Furthermore, if you're not familiar with them, you may want to look up the terms inner product and outer product. They are operations on vectors that, conveniently, map to matrix multiplication between conjugate transposes.
S_David said:Ok then, [tex]\mathbf{h}\mathbf{h}^*[/tex] is the outer product of the two vectors, and [tex]\mathbf{h}^*\mathbf{h}[/tex] is the inner product of them. Right?
Tac-Tics said:Yup!
Usually, the inner product is taken to be a real number, not a 1x1 matrix. But the two ideas are identical. The inner product is well known for it's use is defining orthogonality and angles.
The outer product is less commonly known. It has applications when working with dual vector spaces. It's useful in QM and GR.
A column vector product is a type of mathematical operation that combines two or more column vectors to produce a new vector. It is also known as a dot product or inner product.
The formula for calculating a column vector product is:
c = aT * b = a1b1 + a2b2 + ... + anbn
The results of a column vector product can provide important information about the relationship between the two vectors being multiplied. The magnitude of the product can indicate the strength of the relationship, while the sign can indicate the direction.
Column vector products are commonly used in physics, engineering, and other scientific fields to calculate forces, work, and other important quantities. They can also be used in data analysis and machine learning to find patterns and relationships between variables.
Some properties of column vector products include distributivity, commutativity, and associativity. They also follow the laws of scalar multiplication and can be used to find the angle between two vectors.