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Here is my question. It seems simple but at the same time does not seem to have an obvious answer to me.

Given that you have two vectors [itex]\mathbf{u},\mathbf{v}[/itex].

- They are orthogonal [itex]\mathbf{u}^T\mathbf{v}=0[/itex] by standard dot product definition.
- They have norm one [itex]||\mathbf{u}||=||\mathbf{v}||=1[/itex] by standard dot product definition.
- Define the weighted inner product as [itex]\mathbf{u}^T\left(\begin{matrix}\lambda_1\\&\ddots\\&&\lambda_M\end{matrix}\right)\mathbf{v}[/itex] where [itex]M[/itex] is the number of components. Then the norm according to this inner product is also one for both vectors [itex]\mathbf{u}^T\left(\begin{matrix}\lambda_1\\&\ddots\\&&\lambda_M\end{matrix}\right)\mathbf{u}=

\mathbf{v}^T\left(\begin{matrix}\lambda_1\\&\ddots\\&&\lambda_M\end{matrix}\right)\mathbf{v}=1[/itex]. Notice the dot product is a particular case where the matrix is the identity.- Edit: I forgot to add this [itex]\lambda_1+\cdots+\lambda_M=M[/itex]. It does not make the problem more complicated as it just narrows the possible lambdas.

What is then the maximum inner product (in absolute value) among two vectors satisfying the previous conditions? I.e.

[itex]\operatorname{max}\limits_{\mathbf{u},\mathbf{v}}

\left|

\mathbf{u}^T\left(\begin{matrix}\lambda_1\\&\ddots\\&&\lambda_M\end{matrix}\right)\mathbf{v}

\right|[/itex]

Cheers

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# Maximum inner product between two orthgonal vectors (in standard dot procut))

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