# 15: Matrices

- Page ID
- 106902

Chapter Objectives

- Learn the nomenclature used in linear algebra to describe matrices (rows, columns, triangular matrices, diagonal matrices, trace, transpose, singularity, etc).
- Learn how to add, subtract and multiply matrices.
- Learn the concept of inverse.
- Understand the use of matrices as symmetry operators.
- Understand the concept of orthogonality.
- Understand how to calculate the eigenvalues and normalized eigenvectors of a 2 × 2 matrix.
- Understand the concept of Hermitian matrix

- 15.1: Definitions
- Some types of matrices have special names.

- 15.2: Matrix Addition
- The sum of two matrices A and B (of the same dimensions) is a new matrix of the same dimensions, C = A+ B. The sum is defined by adding entries with the same indices.

- 15.3: Matrix Multiplication
- If A has dimensions m×n and B has dimensions n×p , then the product AB is defined, and has dimensions m×p .

- 15.4: Symmetry Operators
- A symmetry operation, such as a rotation around a symmetry axis or a reflection through a plane, is an operation that, when performed on an object, results in a new orientation of the object that is indistinguishable from the original.

- 15.5: Matrix Inversion
- The inverse of a square matrix A , sometimes called a reciprocal matrix, is a matrix A−1 such that AA−1=I , where I is the identity matrix.

- 15.6: Orthogonal Matrices
- A nonsingular matrix is called orthogonal when its inverse is equal to its transpose.

- 15.7: Eigenvalues and Eigenvectors
- Since square matrices are operators, it should not surprise you that we can determine its eigenvalues and eigenvectors. The eigenvectors are analogous to the eigenfunctions we discussed for quantum mechanics.

- 15.8: Hermitian Matrices
- A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its own conjugate transpose. Hermitian matrices are a generalization of the symmetric real matrices we just talked about, and they also have real eigenvalues, and eigenvectors that form a mutually orthogonal set.