Column Vectors Product

**S_David** · Nov24-09, 02:34 PM

Hello,

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

$LaTeX Code: \\mathbf{h}^*\\mathbf{h}=\\mathbf{h}\\mathbf{h}^*$

where * means complex conjugate transpose?

Thanks in advance

**Tac-Tics** · Nov24-09, 04:23 PM

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

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.

**S_David** · Nov24-09, 04:38 PM

Originally Posted by Tac-Tics

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

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, $LaTeX Code: \\mathbf{h}\\mathbf{h}^*$ is the outer product of the two vectors, and $LaTeX Code: \\mathbf{h}^*\\mathbf{h}$ is the inner product of them. Right?

**Tac-Tics** · Nov24-09, 04:47 PM

Originally Posted by S_David

Ok then, $LaTeX Code: \\mathbf{h}\\mathbf{h}^*$ is the outer product of the two vectors, and $LaTeX Code: \\mathbf{h}^*\\mathbf{h}$ 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.

**S_David** · Nov24-09, 04:56 PM

Originally Posted by Tac-Tics

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