ThmDex – An index of mathematical definitions, results, and conjectures.

Product of real 2-by-2 matrix with its transpose

Formulation 0

Let $A \in \mathbb{R}^{2 \times 2}$ be a D4571: Real matrix.

Then

(1)	\begin{equation} A ' A = \begin{pmatrix} A^2_{1 1} + A^2_{2 1} & A_{1 1} A_{1 2} + A_{2 1} A_{2 2} \\ A_{1 2} A_{1 1} + A_{2 2} A_{2 1} & A^2_{1 2} + A^2_{2 2} \end{pmatrix} \end{equation}
(2)	\begin{equation} A A' = \begin{pmatrix} A^2_{1 1} + A^2_{2 1} & A_{1 2} A_{1 1} + A_{2 2} A_{2 1} \\ A_{1 1} A_{1 2} + A_{2 1} A_{2 2} & A^2_{1 2} + A^2_{2 2} \end{pmatrix} \end{equation}

Formulation 1

Let $A \in \mathbb{R}^{2 \times 2}$ be a D4571: Real matrix.

Then

(1)	\begin{equation} A^T A = \begin{pmatrix} A^2_{1 1} + A^2_{2 1} & A_{1 1} A_{1 2} + A_{2 1} A_{2 2} \\ A_{1 2} A_{1 1} + A_{2 2} A_{2 1} & A^2_{1 2} + A^2_{2 2} \end{pmatrix} \end{equation}
(2)	\begin{equation} A A^T = \begin{pmatrix} A^2_{1 1} + A^2_{2 1} & A_{1 2} A_{1 1} + A_{2 2} A_{2 1} \\ A_{1 1} A_{1 2} + A_{2 1} A_{2 2} & A^2_{1 2} + A^2_{2 2} \end{pmatrix} \end{equation}

Proofs

Proof 0

Let $A \in \mathbb{R}^{2 \times 2}$ be a D4571: Real matrix.

This result is a particular case of R3769: Product of real square matrix with its transpose. $\square$