Matrix Algebra
This section focuses on the content related to matrices within linear algebra, including systems of linear equations, diagonalization of matrices, matrix decomposition, eigenvalue problems, matrix transformations, and more. Information on general vector spaces and linear transformations can be found in the Linear Algebra category. Even if the content is the same, the articles in the Linear Algebra category may be more abstract or complex.
Basic Matrix Algebra
Types of Matrices
- Definition of a Vector
- Definition of a Matrix
- Matrix Operations: Scalar Multiplication, Addition, Multiplication
- Identity Matrix $I, E$
- Zero Matrix $O$
- $1$ Matrix
- Square Matrix
- Triangular Matrix
- Diagonal Matrix $\diag$
- Block Matrix
- 🔒 (24/01/25) Sparse Matrix
Properties of Matrices
- Inverse Matrix $A^{-1}$, Invertible Matrix
- Transpose Matrix $A^{T}$
- Conjugate Transpose Matrix $A^{\ast}$
- Orthogonal Matrix $A^{T} = A^{-1}$
- Projection Matrix $P^{2} = P$
- Positive Definite Matrix $\mathbf{x}^{\ast} A \mathbf{x} \ne 0$
- 🔒 (24/01/29) Square Root of a Matrix $\sqrt{A}$
- 🔒 (24/04/30) Hankel Matrix $H$
Systems of Linear Equations
- Systems of Linear Equations
- Overdetermined and Underdetermined Systems
- Augmented Matrix and Elementary Row Operations
- Gauss-Jordan Elimination
- Homogeneous Systems of Linear Equations
- Elementary Matrices
- Inverse Matrix and Systems of Linear Equations
Determinants
- Determinant $\det$
- 🔒 (23/11/26) Cofactor and Classical Adjoint Matrix $\text{adj} A = C^{T}$
- Proof of Cramer’s Rule
- Derivation of the Determinant of a Vandermonde Matrix
- Derivation of the Determinant of a Tridiagonal Matrix
- 🔒 (23/11/18) Derivation of Sherman-Morrison Formula $\left( A + \mathbf{u} \mathbf{v}^{T} \right)^{-1}$
- 🔒 (23/11/18) Proof of the Matrix Determinant Lemma
Eigenvalue Problems
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Linear Transformations
- Matrix Transformation
- Rotation Transformation
- 🔒 (24/04/22) 3D Rotation Transformation Matrix: Roll, Pitch, Yaw
Matrix Spaces
Numerical Matrix Algebra
Matrix Decomposition
- Eigenvalue Diagonalization of an Invertible Matrix $A = Q^{\ast} \Lambda Q$
- Cholesky Decomposition of a Positive Definite Matrix
- LU Decomposition of an Invertible Matrix $A = LU$
- LDU Decomposition of a Symmetric Matrix $A = LDL^{T}$
- Schur Decomposition of a Square Matrix $A = QTQ^{\ast}$
- Singular Value Decomposition (SVD) of a Matrix
- QR Decomposition of a Matrix
Least Squares Method
References
- Stephen H. Friedberg, Linear Algebra (4th Edition, 2002)
- Kim Sang-dong. (2012). Numerical Matrix Analysis
- Howard Anton, Elementary Linear Algebra: Applications Version (12th Edition, 2019)
All posts
- Permutation Matrix
- PLU Decomposition
- The Spectrum and Decomposition Set of Matrices
- Row Space, Column Space, Null Space
- Matrix Rank, Nullity
- Understanding Ranks and Nullity through Systems of Equations
- Algebraic and Geometric Multiplicities of Eigenvalues
- Similar Matrices Have the Same Eigenvalues
- The Algebraic Multiplicity of Eigenvalues is Greater Than or Equal to Their Geometric Multiplicity
- Eigenvalue Diagonalization of Invertible Matrices
- Singular Value Decomposition of a Matrix
- Proof of the Existence of a Complete Singular Value Decomposition
- Schur Decomposition of Square Matrices
- Eigenvalue Diagonalization of Hermitian Matrices: Proof of Spectral Theory
- LU Decomposition of Invertible Matrices
- Decomposition of Symmetric Matrices into LDU
- Cholesky Decomposition of Positive Definite Matrices
- Uniqueness Proof of Cholesky Decomposition
- Matrix Algebra: Projections
- Matrix Projection in Linear Algebra
- Least Squares Method
- Matrix QR Decomposition
- Cholesky Decomposition for Least Squares Method
- QR Decomposition for Least Squares Method
- Singular Value Decomposition for Least Squares
- Laplace Expansion
- Proof of Cramer's Rule
- Derivation of the Determinant of the Vandermonde Matrix
- Tri-diagonal Matrix Determinant Derivation
- Strassen Algorithm Proof
- Generalization of the Ellipse: Ellipsoid
- Operations and Notation Table of Vectors and Matrices
- Definition of Vectors
- Matrix Definitions
- Matrix Operations: Scalar Multiplication, Addition, and Multiplication
- Square Matrix
- Diagonal Matrix
- Identity Matrix, Unit Matrix
- Transpose Matrix
- Inverse Matrix, Reversible Matrix
- Conditions for a Matrix Being Invertible
- Symmetric Matrices, Skew-Symmetric Matrices
- Conjugate Transpose Matrix
- Matrix Inner Product
- Orthogonal Matrix
- Properties of Orthogonal Matrices
- Trace
- Equivalence Conditions for Orthogonal Matrices
- Hermitian Matrix
- The eigenvalues of a Hermitian matrix are always real
- The eigenvectors of distinct eigenvalues of Hermitian matrices are orthogonal.
- Unitary Matrix
- Simultaneous Linear Equations
- Augmented Matrices and Elementary Row Operations
- Determinants
- Properties of Determinants
- Definite matrix
- Eigenvalues and Eigenvectors
- Matrix Similarity
- Matrix Transformation
- Gaussian-Jordan Elimination
- Simultaneous Homogeneous Linear Equations
- Basic Matrix
- Inverse Matrices and Systems of Linear Equations
- Basis of Row Space, Column Space, and Null Space
- Pseudo Inverse Matrix
- Rotational Transformation
- Frobenius Norm
- Supersaturated and Undersaturated Systems
- Fundamental Spaces of Matrices
- Properties of Full Rank Matrices
- Perron-Frobenius Theorem
- Triangular Matrix
- Power Matrix
- Definition of Spectral Radius
- Block Matrices
- Row-wise and Column-wise Scalar Multiplication of Matrix
- Orthogonal Triangular Matrices are Nilpotent
- Matrix
- Zero Matrix
- Direct Sum of Matrices
- Kronecker Product of Matrices
- Hadamard Product of Matrices
- Proof of the Matrix Determinant Lemma
- Derivation of the Sherman-Morrison Formula
- Eigenvalues and Eigenvectors
- Determinant of a Triangular Matrix
- Sparse Matrices
- Square Root Matrix
- Finding the Inverse Matrix Using Gaussian Elimination Algorithm
- Inverse Matrix of X^T X: Necessary and Sufficient Conditions
- 3D Rotation Transformation Matrix: Roll, Pitch, Yaw
- Hankel Matrix
- Toeplitz Matrices are Hermitian Matrices