Matrix theory. Matrix, vector space, Linear Transformation,
Congruence, Similarity, Eigenvalue, eigenvector, characteristic equation,
Smith normal form, invariant factors, elementary divisors, Characteristic
matrix, similarity invariants, minimum polynomial, companion matrix, canonical
form, Skew-symmetric matrix, Direct Sum, Hermitian matrices, Skew-Hermitian
matrix, Row equivalence, Row Canonical Form, Gramian Matrix, Unitary matrix,
Unitary transformation.

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# Matrix theory

Select from the following:
Preliminary definitions, conventions and notations. n-vectors,
n-space, dot products, m-dot products

Matrices, Operations on matrices, Algebraic laws obeyed by matrices

Matrix products by partitioning: block
multiplication of matrices

Types of Matrices, Triangular matrix, Diagonal matrix, Scalar matrix,
Identity matrix, Symmetric matrix, Skew-symmetric matrix, Direct Sum,
Transpose of a matrix

Complex numbers, conjugate complex numbers, complex conjugate matrices

Hermitian matrices, Skew-Hermitian matrix, Hermitian conjugate of
a matrix

Determinant, Minor, Cofactor, Evaluation of a
determinant by cofactors

Elementary operations on matrices, Inverse operations, Equivalent
matrices, Row equivalence, Row Canonical Form, Elementary row
and column operations effected through multiplication by
elementary matrices, Reduction to canonical form

Vector spaces and subspaces, linearly dependent and independent
sets of vectors, space spanned by a set of vectors, basis of a
vector space, sum and intersection space of two vector spaces,
coordinate systems in vector spaces, changes in coordinates due to
change in basis

Row space of a matrix, Column space

Some theorems

Adjoint of a matrix, inverse of a matrix

Functions, mappings, maps, transformations, operators

Linear Transformations, singular and non-singular transformations

Ways of viewing Y = AX

Matrix theorems

Expansions of matrix-vector products

Effect of multiplying a matrix by a diagonal matrix

Products of the type XTX, XXT, XTDX ATA and AAT where X is a vector,
A is a matrix, and D is a diagonal matrix.

An interpretation of the product of two matrices

The solution set of the linear system AX = 0 is a vector space

Systems of linear equations. Equivalence, independence, dependence, consistency

Systems of linear equations, matrix solution, augmented
matrix, homogeneous and non-homogeneous systems, Cramer's rule, null
space

Solution of a consistent system of linear equations

Intuitive interpretation of the solution sets of AX = B and AX = 0

Technique for solving underdetermined systems of linear
equations

Vectors over real n-space, Orthogonal vectors and spaces, Triangle
Inequality, Schwarz Inequality, Gram-Schmidt orthogonalization
process, Gramian Matrix

Vectors over complex n-space, Orthogonal vectors, Triangle
Inequality, Schwarz Inequality, Gram-Schmidt orthogonalization
process, Gramian Matrix, Unitary matrix, Unitary transformation

Congruence, Congruent Transformation, Symmetric matrices,
Skew-symmetric matrices, Hermitian matrices, Skew-Hermitian matrices

Bilinear forms, Reduction to canonical form, Cogredient Transformations,
Contragredient transformations

Quadratic forms, Reduction to canonical form, Lagrange's Reduction,
Definite and semi-definite forms, Regular quadratic form

Hermitian forms, Conjunctive Hermitian matrices, definite and
semi-definite forms

Eigenvalues and eigenvectors, characteristic equation, characteristic
polynomial, characteristic root

Similarity, Similar matrices, Orthogonal similarity, Real quadratic
forms, Hermitian matrices, Normal matrices

Affine transformations

Eigenvectors and their meaning

A linear point transformation Y = AX viewed as occurring in three steps

Ways of viewing a non-singular linear transformation Y = AX

Linear transformation Y = AX viewed as a product of rotations and elongations /contractions in mutually orthogonal directions

Divisors, factors and multiples of integers; Common divisor,
Greatest common divisor, Common multiple, Least common multiple,
Division Algorithm, Euclid's algorithm, Unique Factorization Theorem

Polynomials over a field, Polynomial domain, Quotients of polynomials,
Remainder theorem, Greatest common divisor, Unique Factorization
Theorem

Lambda matrices, matrix polynomials, division of lambda-matrices,
remainder theorem, scalar matrix polynomials, Cayley-Hamilton theorem

Smith normal form, invariant factors, elementary divisors

Characteristic matrix, similarity invariants, minimum polynomial,
companion matrix, non-derogatory matrices

Canonical forms under similarity; Rational, Jacobson and Jordan
canonical forms; hypercompanion matrix

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