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
Rank of a matrix, 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 X'X, XX', X'DX A'A and AA' where X is a vector,
A is a matrix, D is a diagonal matrix, and prime is transpose.
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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