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A set of vector is said to form a linear vector space V

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Basis vector
The inner product is defined by

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some properties of inner product

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Some useful inequalities:

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8.2 Linear operators
The action of operator A is independent of any basis or coordinate system.

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Properties of linear operators:

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8.4 Basic matrix algebra
Matrix addition
Multiplication by a scalar

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Multiplication of matrices

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Functions of matrices
The transpose of a matrix

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For a complex matrix

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If the basis is not orthonormal
The trace of a matrix

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8.9 The determinant of a matrix

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Ex: Matrix A is 3×3, for three 3-D vectors
Properties of determinants

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Ex: Evaluate the determinant
(4)-(2) put (4)
(2)+(3) put (3)

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8.10 The inverse of a matrix
The elements of the inverse matrix are

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Useful properties:

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8.12 Special types of square matrix
Diagonal matrices

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Lower triangular matrix
Upper triangular matrix
Symmetric and antisymmetric matrices

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Ex: If A is N×N antisymmetric matrix, show that |A|=0 if N is odd.
Orthogonal matrices
Hermitian and anti-Hermitian matrices

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Unitary matrices
Normal matrices

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8.13 Eigenvectors and eigenvalues
Ex: A non-singular matrix A has eigenvalues λi , and eigenvectors xi. Find the eigenvalues and eigenvectors of the inverse matrix A-1.

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Eigenvectors and eigenvalues of a normal matrix

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An eigenvalue corresponding to two or more different eigenvectors is said to be degenerate.

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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

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Ex: Prove that the eigenvectors corresponding to different eigenvalues of an Hermitian matrix are orthogonal.
anti-Hermitian matrix

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Unitary matrix
Simultaneous eigenvectors

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8.14 Determination of eigenvalues and eigenvectors
Ex: Find the eigenvalues and normalized eigenvectors of the real symmetric matrix

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Sol:

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Degenerate eigenvalues

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8.15 Change of basis and similarity transformations

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Similarity transformations
similarity transformation
Properties of the linear operator under two basis

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If S is a unitary matrix:

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8.16 Diagonalization of matrices

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For normal matrices (Hermitian, anti-Hermitian and unitary) the N eigenvectors are linear independent.

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Ex: Prove the trace formula

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8.17 Quadratic and Hermitian forms

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The stationary properties of the eigenvectors.
What are the vector that form a maximum or minimum of

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Ex: Show that if is stationary then is an eigenvector of and
is equal to the corresponding eigenvalues.

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8.18 Simultaneous linear equation

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Ex: Use Cramer’s rule to solve the set of the simultaneous equation.