The Exercise

Take an inner product space $V$ of finite dimension $d$, over the real or the complex numbers, with an orthonormal basis $|b_1\rangle, \dots, |b_d\rangle$, and pick two of its vectors, $|b_j\rangle$ and $|b_k\rangle$. Their outer product $|b_j\rangle\langle b_k|$ is an operator on $V$. The exercise asks for its matrix in the basis $|b_i\rangle$.

Outer Products and Matrices

The outer product $|x\rangle\langle y|$ of two vectors $|x\rangle$ and $|y\rangle$ of $V$ is the operator on $V$ that sends each vector $|z\rangle$ to $|x\rangle$ multiplied by the inner product $\langle y|z\rangle$: $$\bigl(|x\rangle\langle y|\bigr)\,|z\rangle = \langle y|z\rangle\,|x\rangle .$$(1)

The matrix $[A]$ of an operator $A$ on $V$ in the orthonormal basis $|b_i\rangle$ is the $d \times d$ matrix whose entry in row $i$ and column $l$ is $$[A]_{il} = \langle b_i|A|b_l\rangle .$$(2)

For indices $j$ and $k$ from $1$ to $d$, the matrix unit $E_{jk}$ is the $d \times d$ matrix with $1$ in row $j$ and column $k$ and $0$ in every other entry: $$(E_{jk})_{il} = \delta_{ij}\,\delta_{kl} .$$(3)

The Claim

What Is Checked

The claim is stated over an orthonormal basis of $V$ and the constructions above: the outer product (1), the matrix of an operator in that basis (2) and the matrix unit (3). What the library proves about them is the equation (4).