Two Linear Algebra Proofs - Hermitian and Transpose: Quantum Mechanics (Griffiths)

Problem A.11 from D.J. Griffiths' "Introduction to Quantum Mechanics: 2nd Edition" asks the student to prove a few properties from linear algebra. The properties aren't all that difficult to prove, but give some students trouble because these properties are generally used by students to prove other relationships in linear algebra.

The first property: The transpose of a product is the product of the transposes in reverse order...

$$\widetilde{\mathbf{ST}}=\widetilde{\mathbf{T}}\widetilde{\mathbf{S}}$$

$$\widetilde{\mathbf{ST}}$$

$$=(ST)_{ji}$$

$$=\sum_{k=1}^{n}S_{jk}T_{ki}$$

$$=\sum_{k=1}^{n}\widetilde{S}_{kj}\widetilde{T}_{ik}$$

$$=\sum_{k=1}^{n}\widetilde{T}_{ik}\widetilde{S}_{kj}$$

$$=\mathbf{\widetilde{T}}\mathbf{\widetilde{S}}$$

QED

The second property: The Hermitian conjugate of a product is the product of the Hermitian conjugates in reverse order...

$$(\mathbf{ST})^{\dagger }=\mathbf{T}^{\dagger }\mathbf{S}^{\dagger }$$

$$(\mathbf{ST})^{\dagger }$$

$$=(\widetilde{\mathbf{ST}})^{*}$$

$$=(\mathbf{\widetilde{T}}\mathbf{\widetilde{S}})^{*}$$

$$=\mathbf{\widetilde{T}}^{*}\mathbf{\widetilde{S}}^{*}$$

$$=\mathbf{T}^{\dagger }\mathbf{S}^{\dagger }$$

QED