# Determine if vector space of all 2 x 2 matrices is a inner product space

This is a question from textbook.

Determine if the vector space of all $2 \times 2$ matrices is a inner product. Let $A$ and $B$ be $2\times 2$ matrices then $\langle A, B \rangle = a_1b_1 + a_2b_2 + a_3b_3 + a_4b_4$.

My understanding is that this is not an inner product because it does not satisfy P4 $\langle v, w \rangle \geq 0$ for all $v$ and $w$. Suppose we let $A =\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 0 \\ 0 & 0 \end{bmatrix}$ then $\langle A, B \rangle = -1$ therefore it is not an inner product. However, the text solution states that it is an inner product. What am I misunderstanding here? I apologize for the bad syntax.

• The inequality you are trying to remember is $(u,u)\geq 0$, the same element multiplied by itself. Aug 5, 2017 at 12:39
• Thank you, I totally missed that. Aug 5, 2017 at 12:40
• That operation is going to be an inner product. The space of $2\times2$ matrices is the same as $\mathbb{R}^4$ by sending $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ to $(a,b,c,d)$. That formula is just the standard inner product on $\mathbb{R}^4$. Aug 5, 2017 at 12:44

Let $A$ be a non-zero matrix with entries $[A]_{ij} = a_{ij}$ Then $\langle A, A \rangle = \langle \begin{pmatrix} a_{11} & a_{12} \\a_{21} & a_{22}\end{pmatrix}, \begin{pmatrix} a_{11} & a_{12} \\a_{21} & a_{22}\end{pmatrix} \rangle = a_{11}^2 + a_{12}^2 + a_{21}^2 + a_{22}^2 > 0$
To understand why we require positieve definiteness, in any kind of inner vector space, either real or complex, you must remember that the norm must always be a positive number (as this represents a distance, and distances can't be negative, can they?). Since $\Vert v \Vert = \sqrt{\langle v,v \rangle}$, it makes sense to ask positive definiteness, or the number under the square root is possibly negative, something we don't want.