Let $\pmb{b}$ and $\pmb{x}$ be two vectors in $\mathbb{R}^n$, then: $$(\pmb{b}^\mathrm{T}\pmb{x})^2=(\pmb{b}^\mathrm{T}\pmb{x})(\pmb{b}^\mathrm{T}\pmb{x})=(\pmb{b}^\mathrm{T}\pmb{x})(\pmb{x}^\mathrm{T}\pmb{b})$$ Now, is $(\pmb{b}^\mathrm{T}\pmb{x})(\pmb{x}^\mathrm{T}\pmb{b})=\pmb{b}^\mathrm{T}(\pmb{x}\pmb{x}^T)\pmb{b}$ true? I cannot find a counterexample, but I wonder what property it is.
2026-02-23 10:50:36.1771843836
Is $(\pmb{b}^\mathrm{T}\pmb{x})^2=\pmb{b}^\mathrm{T}(\pmb{x}\pmb{x}^T)\pmb{b}$ true?
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Yes, this is true. It is simply a consequence of the associativity of matrix multiplication:
$$ (AB)(CD) = A(BC)D $$
Here, think of $\mathbf{b}$ and $\mathbf{x}$ as $n \times 1$ matrices, and take $A = \mathbf{b}^T$, $B = \mathbf{x}$, $C = \mathbf{x}^T$, and $D = \mathbf{b}$.