Features

Clear and concise explanations

Difficult concepts are explained in simple terms

Illustrated with graphs and diagrams

Table of Contents

I. Linear equations

System of linear equations

Determinant

Minor

Cauchy-Binet formula

Cramer's rule

Gaussian elimination

Gauss-Jordan elimination

Strassen algorithm

II. Matrices

Matrix addition

Matrix multiplication

Basis transformation matrix

Characteristic polynomial, Characteristic Equation

Trace

Eigenvalue, eigenvector and eigenspace

Cayley-Hamilton theorem

Spread of a matrix

Symbolic Computation of Matrix Eigenvalues

Jordan normal form

Rank

Matrix inversion,

Pseudoinverse

Adjugate

Transpose

Dot product

Symmetric matrix

Matrix congruence

Congruence relation

Orthogonal matrix

Skew-symmetric matrix

Conjugate transpose

Unitary matrix

Hermitian matrix, Antihermitian

Positive definite: matrix, function, bilinear form

Identity matrix

Pfaffian

Projection

Diagonal matrix, main diagonal

Diagonalizable matrix

Similar matrix

Tridiagonal matrix

Hessenberg matrix

Triangular matrix

Spectral theorem

Stochastic matrix

Toeplitz matrix

Circulant matrix

Hankel matrix

Vandermonde matrix

Block matrix

(0,1)-matrix

Normal Matrix

Sparse matrix

Woodbury matrix identity

Perron-Frobenius theorem

List of Matrices

III. Matrix decompositions

Block LU Decomposition

Cholesky decomposition

LU decomposition

QR decomposition

Spectral theorem

Singular value decomposition

Schur decomposition

Schur complement

IV. Computations

Transformation Matrix

Householder transformation

Least squares, linear least squares

Gram-Schmidt process

V. Vectors

Unit Vector

Pseudovector

Normal Vector

Tangential and Normal Components

Scalar multiplication

Linear combination

Linear span

Linear independence

Basis

VI. Vector spaces

Basis=Hamel basis

Dimension theorem for vector spaces=Hamel dimension

Examples of vector spaces

Linear map

Galilean transformation, Lorentz transformation

Row and Column space

Null space

Rank-nullity theorem

Dual space

Linear function

Linear functional

Orthogonality

Orthogonal complement

Orthogonal projection

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