# 15: Matrices

• • Contributed by Marcia Levitus

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.
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.
• 15.9: Problems