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I have the following expression:

$(\mathbf{x} - \mathbf{\Psi a})^{T}(\mathbf{x} - \mathbf{\Psi a})$

I would like to develop the expression and gather the terms:

$\mathbf{x}^{T}\mathbf{x} - \mathbf{x}^{T}\mathbf{\Psi a} - (\mathbf{\Psi a})^{T}\mathbf{x} + (\mathbf{\Psi a})^{T}\mathbf{\Psi a}$

$ = \mathbf{x}^{T}\mathbf{x} -2\mathbf{x}^{T}\mathbf{\Psi a} + \mathbf{a^{T}\Psi^{T}\Psi a}$

Why is the last expression is true?

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  • $\begingroup$ Is it in case that $\Psi$ is semi-positive definite? $\endgroup$
    – Jose Ramon
    Commented Sep 10, 2020 at 18:17
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    $\begingroup$ The transpose of a scalar is the same scalar. And $x^T \Psi a$ is a scalar and equal to $a^T \Psi^T x$. $\endgroup$
    – Mark Viola
    Commented Sep 10, 2020 at 18:18
  • $\begingroup$ ok that was easy and make sense. Thanks for the help! $\endgroup$
    – Jose Ramon
    Commented Sep 10, 2020 at 18:20
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    $\begingroup$ You're welcome. I could post it as a solution. $\endgroup$
    – Mark Viola
    Commented Sep 10, 2020 at 18:20
  • $\begingroup$ Go for it! I will accept it. $\endgroup$
    – Jose Ramon
    Commented Sep 10, 2020 at 18:21

1 Answer 1

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The transpose of a scalar is the same scalar.

And note that $x^T \Psi a$ is indeed a scalar.

So, it is equal to its transpose and we have

$$x^T\Psi a= (x^T\Psi a)^T=(\Psi a)^T x=a^T \Psi^T x$$

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