Matrices

• What Are Left and Right Eigenvalues?
  For an ordinary real or complex square matrix, left and right eigenvalues are the same; the associated eigenvectors satisfy different equations.
• How to Find the Inverse of a Circulant Matrix
  Find a circulant matrix inverse by evaluating its symbol polynomial at the nth roots of unity, then applying the inverse discrete Fourier transform.
• Difference Between the Dimension and Rank of a Matrix
  Learn why matrix size is not the same as rank, how rank is defined as a space dimension, and how to calculate both with an example.
• How to Find the Dimension of the Kernel of a Square Matrix
  For an n × n matrix, the kernel has dimension n minus the matrix rank. Row-reduction gives the rank and therefore the nullity.
• How to Compute the Semisimple Part of a Matrix
  Compute a matrix’s semisimple part by replacing each Jordan block with its eigenvalue times the identity, or by evaluating a suitable polynomial.
• How to Find the Rank of a Matrix A with A² = A
  If A² = A, then A is idempotent. Learn why its rank can range from 0 to n, how to compute it from the entries, and why rank equals trace.
• What Is an Example of a Degenerate Bilinear Map?
  A bilinear map is degenerate when a nonzero vector pairs to zero with every vector; see matrix tests and concrete examples in its space.
• How to Find a Gram Matrix Determinant Without Special Methods
  Find a Gram matrix determinant using ordinary determinant rules, with a worked 3×3 example and a formula for symmetric matrices.
• What Is the Determinant of a Scalar Times a Matrix?
  For an n by n matrix A and scalar c, det(cA) equals c^n det(A), not simply c det(A).
• Why Do Arithmetic-Progression Columns Give a Zero Determinant?
  If every column is an arithmetic progression, each lies in the span of two fixed vectors, so an n×n determinant is zero for n at least 3.
• How to Convert MATLAB Arrays to LaTeX Matrices
  Convert a numeric MATLAB array into LaTeX matrix code with a short reusable function, control numeric precision, and compile the result.
• What Conditions Make the Product of Two Matrices Zero?
  For compatible matrices, AB = 0 exactly when the image of B lies in the kernel of A; this also implies a rank bound.
• Why Do Rotation Matrices About the y-Axis Have Different Signs?
  For column vectors in a right-handed system, a positive active rotation about y has a specific sign pattern; the opposite is its inverse.
• How to Use Variation of Parameters for a 3×3 System
  Use a fundamental matrix to solve a nonhomogeneous 3×3 linear ODE system by variation of parameters, with a worked example.
• Examples of 3×3 Matrices With Common Properties
  Browse verified 3×3 matrix examples for symmetry, rank, invertibility, orthogonality, idempotence, nilpotence, and noncommutativity.
• What Is the Relationship Between Eigendecomposition and SVD?
  The SVD is linked to the eigendecompositions of A* A and A A*: their eigenvectors give singular vectors and their eigenvalues are squared singular values.