Chapter 2: Operations with Matrices
- §1. Matrix Addition
- §2. Scalar Multiplication
- §3. Matrix Multiplication
- §4. Transpose
- §5. Hermitian Adjoint
- §6. Trace
- §7. Determinants
- §8. Inverses
- §9. Bra-Ket Notation
Scalar Multiplication
A matrix can be multiplied by a scalar, in which case each element of the matrix is multiplied by the scalar. In components, $$C_{ij}=\lambda A_{ij}$$ where $\lambda$ is a scalar, that is, a complex number. For example, if $$A = \left(\begin{array}{cc} a&b\\ c&d\\ \end{array} \right),$$ then $$3A=3\cdot \left(\begin{array}{cc} a&b\\ c&d\\ \end{array} \right) = \left(\begin{array}{cc} 3a&3b\\ 3c&3d\\ \end{array} \right).$$
Try it for yourself by computing $$i\cdot \left(\begin{array}{cc} 1&i\\ -2i&3\\ \end{array} \right).$$