This is a set of lecture notes on matrix algebra. Use these lectures for self-study or as a complement to your textbook.
How to add two matrices together, definition and properties of addition
Matrices, their characteristics, introduction to some special matrices
Obtained by multiplying matrices by scalars, and by adding them together
Multiplication of a matrix by a scalar
How to multiply a matrix by a scalar, definition and properties of scalar multiplication
It plays in matrix multiplication the same role played by 1 in the multiplication of numbers
How to multiply two matrices, definition and properties of multiplication
Addition, scalar multiplication and multiplication of block matrices can be performed on their blocks
A matrix that has been partitioned into smaller submatrices
This lecture introduces one of the central concepts in matrix algebra
Sets of vectors that are closed with respect to taking linear combinations
A set of linearly independent vectors that span the linear space
The linear space generated by taking linear combinations of a set of vectors
A basis made up of vectors that have all entries equal to zero except one
The number of elements of any one of the bases of the linear space
Two subspaces are complementary if their direct sum equals the whole space
The sum of two subspaces whose intersection contains only the zero vector
The dimension of the linear space spanned by the columns or rows of the matrix
Matrix product and linear combinations
Multiplying matrices is equivalent to taking linear combinations of their rows and columns
Multivariate generalization of the concept of reciprocal of a number
This lecture note presents some useful facts about the rank of the product of two matrices
A device that helps to invert and factorize block matrices
Formulae for computing how changes in a matrix affect its inverse
Functions that preserve vector addition and scalar multiplication
Vectors containing the coefficients of the representation in terms of a basis
Linear transformations that map a space into itself
Each linear map is associated to a unique matrix that transforms coordinates
The set of vectors belonging to the domain that are mapped into the zero vector
The composition of two linear transformations is itself linear
Surjective, injective and bijective maps
Learn how to classify maps based on their kernel and range
The subset of the codomain formed by all the values taken by the map
Learn what happens to coordinate vectors when you switch to a different basis
The dimension of the domain of a linear map equals the sum of the dimensions of its kernel and range
The matrix of a linear operator which projects vectors onto a subspace
Equivalent systems of equations
Systems of linear equations having the same set of solutions
Systems of linear equations can be written compactly and easily studied with matrices
A compact way to represent systems of linear equations
Elementary operations used in matrix algebra to transform a linear system into an equivalent system
The main algorithm used to reduce linear systems to row echelon form
Systems of linear equations having this form can be easily solved with the back-substitution algorithm
The standard algorithm used to transform linear systems to reduced row echelon form
Echelon form in which the basic columns are vectors of the standard basis
A system of equations in which the vector of constants is non-zero
A system of equations in which the vector of constants is zero
Operations that allow us to transform a linear system arranged horizontally into an equivalent system
A matrix that has all entries below (or above) the main diagonal equal to zero
A matrix used to perform multiple interchanges of rows and columns
A matrix whose off-diagonal entries are all equal to zero
A matrix obtained by performing an elementary operation on an identity matrix
How to write a matrix as a product of a lower and an upper triangular matrix
How elementary row operations generate equivalence classes
Taking both the transpose and the complex conjugate of a matrix is very common in matrix algebra
Basic facts and definitions about matrices whose entries are complex numbers
The norm of a vector generalizes the concept of length to abstract spaces
A generalization of the concept of dot product to abstract vector spaces
A procedure used in matrix algebra to create sets of orthonormal vectors
A basis whose vectors are orthogonal and have unit norm
A=QR where Q has orthonormal columns and R is upper triangular
A complex matrix whose columns form an orthonormal set
A special case of oblique projection that gives the closest vector in the subspace
The subspace formed by all the vectors that are orthogonal to a given set
A unitary matrix often used to transform another matrix into a simpler one
The ranges and kernels of a matrix and its transpose are pairwise orthogonal complements
An orthogonal matrix that can be used to perform equivalent transformations
A number telling us how the associated linear transformation scales volumes
A concept that pops up in the definition of determinant of a matrix
Discover several properties enjoyed by the determinant of a matrix
Determinants of elementary matrices
Determinants of elementary matrices enjoy some special properties
Laplace expansion, minors and cofactors
A formula for easily computing the determinant of a matrix
Rules about the determinants of block matrices are very useful
The trace of a matrix is the sum of the entries on its main diagonal
Dividend equals divisor times quotient plus remainder, achieved with the Division Algorithm
These lecture notes summarize some facts about polynomials that are important in matrix algebra
The greatest common divisor of polynomials has properties similar to the gcd of integers
The polynomial whose roots are the eigenvalues of a matrix
Linear transformations scale up or down the sides of certain parallelograms but do not change their angles
Linear independence of eigenvectors
Eigenvectors corresponding to distinct eigenvalues are linearly independent
Algebraic and geometric multiplicities
The multiplicity of a repeated eigenvalue and the dimension of its eigenspace
Similar matrices have the same rank, trace, determinant and eigenvalues
Eigenvalues and eigenvectors possess several useful properties that are also easy to derive
Any matrix is unitarily similar to an upper triangular matrix
Transformation of a matrix into another similar matrix that is diagonal
A full-rank matrix whose eigenvalues are all strictly positive
A matrix that commutes with its conjugate transpose and is unitarily diagonalizable
Write a matrix as a product of a unitary, a diagonal and another unitary matrix
This lecture explains how to factorize a matrix into a lower triangular matrix and its conjugate transpose
A subspace that is mapped into itself by a linear operator
| Decomposition | Conditions on A | Properties of matrices |
|---|---|---|
| A = LU | No row interchanges for REF | L lower triangular, U upper triangular |
| PA = LU | No conditions | P permutation, L and U lower and upper triangular |
| A = QR | Full-rank | Q unitary, R upper with diagonal entries > 0 |
| A = PDP-1 (Diagonalization) | No defective eigenvalues | D diagonal, P invertible |
| A = QTQ* (Schur) | No conditions | T upper triangular, Q unitary |
| A = LL* (Choleski) | Positive definite | L lower triangular with diagonal entries > 0 |
| A = USV* (Singular Value) | No conditions | U and V unitary, S diagonal with entries >= 0 |
| A = PJP-1 (Jordan) | No conditions | J in Jordan form, P invertible |
Range null-space decomposition
A certain power of a matrix can be used to decompose a space of vectors
Discover what happens to the null and column spaces of a matrix when you raise it to integer powers
If you transform the characteristic polynomial into a matrix polynomial, you get the zero matrix
Matrix powers can be used to construct polynomials, similarly to the scalar case
These lecture notes explain the most important application of the minimal polynomial
The annihilating polynomial having the lowest possible degree
Nilpotent matrices generate strings of linearly independent vectors
A matrix that becomes equal to the zero matrix if raised to a sufficiently high power
A string of generalized eigenvectors ending with an ordinary eigenvector
A vector that can be used to complete a basis of eigenvectors when the matrix is defective
Any matrix is similar to an almost diagonal matrix, said to be in Jordan form
Properties of Kronecker products
This lecture note presents several uselful properties of the Kronecker product
A big matrix that contains all the products of the entries of two matrices
A permutation matrix used to transpose vectorizations and commute Kronecker products
An operator that transforms any matrix into a column vector
How to apply scalar functions such as the exponential to square matrices
A workhorse of engineering and a useful application of matrix algebra
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