Can Column Vectors be Multiplied?

[EngWiPy]

Hello,

Suppose that \mathbf{h} is an N\times 1 column vector. Can we say that:

\mathbf{h}^*\mathbf{h}=\mathbf{h}\mathbf{h}^*

where * means complex conjugate transpose?

Thanks in advance

 

[Tac-Tics]

Try it out for yourself. Take \mathbf{h} = [1, 0]^*.

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.

 

[EngWiPy]

Try it out for yourself. Take \mathbf{h} = [1, 0]^*.

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.

Ok then, \mathbf{h}\mathbf{h}^* is the outer product of the two vectors, and \mathbf{h}^*\mathbf{h} is the inner product of them. Right?

 

[Tac-Tics]

Ok then, \mathbf{h}\mathbf{h}^* is the outer product of the two vectors, and \mathbf{h}^*\mathbf{h} is the inner product of them. Right?

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.

 

[EngWiPy]

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.

Ok thank you Tac-Tics.

Best regards