8.1 Vector spaces A set of vector is said to form a linear vector space V Chapter 8 Matrices and vector spaces
Basis vector The inner product is defined by
Chapter 8 Matrices and vector spaces some properties of inner product
Chapter 8 Matrices and vector spaces
Some useful inequalities: Chapter 8 Matrices and vector spaces
8.2 Linear operators The action of operator A is independent of any basis or coordinate system.
Chapter 8 Matrices and vector spaces Properties of linear operators:
Chapter 8 Matrices and vector spaces 8.3 Matrices
Chapter 8 Matrices and vector spaces 8.4 Basic matrix algebra Matrix addition Multiplication by a scalar
Chapter 8 Matrices and vector spaces Multiplication of matrices
Chapter 8 Matrices and vector spaces
Functions of matrices The transpose of a matrix
Chapter 8 Matrices and vector spaces For a complex matrix
Chapter 8 Matrices and vector spaces If the basis is not orthonormal The trace of a matrix
Chapter 8 Matrices and vector spaces 8.9 The determinant of a matrix
Chapter 8 Matrices and vector spaces Ex: Matrix A is 3×3, for three 3-D vectors Properties of determinants
Chapter 8 Matrices and vector spaces
Ex: Evaluate the determinant (2)+(3) put (3) (4)-(2) put (4)
The elements of the inverse matrix are Chapter 8 Matrices and vector spaces 8.10 The inverse of a matrix
Chapter 8 Matrices and vector spaces Useful properties:
Chapter 8 Matrices and vector spaces 8.12 Special types of square matrix Diagonal matrices
Chapter 8 Matrices and vector spaces Lower triangular matrix Upper triangular matrix Symmetric and antisymmetric matrices
Chapter 8 Matrices and vector spaces Ex: If A is N×N antisymmetric matrix, show that |A|=0 if N is odd. Orthogonal matrices Hermitian and anti-Hermitian matrices
Chapter 8 Matrices and vector spaces Unitary matrices Normal matrices
Chapter 8 Matrices and vector spaces 8.13 Eigenvectors and eigenvalues Ex: A non-singular matrix A has eigenvalues λ i, and eigenvectors x i. Find the eigenvalues and eigenvectors of the inverse matrix A -1.
Chapter 8 Matrices and vector spaces Eigenvectors and eigenvalues of a normal matrix
Chapter 8 Matrices and vector spaces An eigenvalue corresponding to two or more different eigenvectors is said to be degenerate.
Chapter 8 Matrices and vector spaces
Ex: Show that a normal matrix can be written in terms of its eigenvalues and orthogonal eigenvectors as Eigenvectors and eigenvalues of Hermitian and anti-Hermitian matrices
Chapter 8 Matrices and vector spaces Ex: Prove that the eigenvectors corresponding to different eigenvalues of an Hermitian matrix are orthogonal. anti-Hermitian matrix
Chapter 8 Matrices and vector spaces Unitary matrix Simultaneous eigenvectors
Chapter 8 Matrices and vector spaces
8.14 Determination of eigenvalues and eigenvectors Ex: Find the eigenvalues and normalized eigenvectors of the real symmetric matrix
Chapter 8 Matrices and vector spaces Sol:
Chapter 8 Matrices and vector spaces
Degenerate eigenvalues
Chapter 8 Matrices and vector spaces 8.15 Change of basis and similarity transformations
Chapter 8 Matrices and vector spaces Similarity transformations similarity transformation Properties of the linear operator under two basis
Chapter 8 Matrices and vector spaces If S is a unitary matrix:
Chapter 8 Matrices and vector spaces 8.16 Diagonalization of matrices
Chapter 8 Matrices and vector spaces For normal matrices (Hermitian, anti-Hermitian and unitary) the N eigenvectors are linear independent.
Chapter 8 Matrices and vector spaces Ex: Prove the trace formula
Chapter 8 Matrices and vector spaces 8.17 Quadratic and Hermitian forms
Chapter 8 Matrices and vector spaces
The stationary properties of the eigenvectors. What are the vector that form a maximum or minimum of
Chapter 8 Matrices and vector spaces Ex: Show that if is stationary then is an eigenvector of and is equal to the corresponding eigenvalues.
Chapter 8 Matrices and vector spaces 8.18 Simultaneous linear equation
Chapter 8 Matrices and vector spaces
Ex: Use Cramer’s rule to solve the set of the simultaneous equation.