Linear Algebra Trivia Questions
- What is the result of the Rank-Nullity Theorem when applied to a linear transformation $L: V \to W$?
- What is the solution to the matrix differential equation dx/dt = Ax with initial condition x(0) = x₀?
- What is the specific matrix representation of the derivative operator on the space of polynomials of degree at most 2 in the basis {1, x, x^2}?
- What is the term for a set of vectors where every vector is orthogonal to every other vector and each has a magnitude of one?
- What is the term for a square matrix A such that A² = A?
- What is the term for a square matrix that does not have a full set of linearly independent eigenvectors?
- What is the term for a square matrix where all entries are non-negative and every row and column sums to exactly one?
- What is the term for a nonzero vector that only changes by a scalar factor when a linear transformation is applied to it?
- What is the term for the set of all eigenvalues of a matrix?
- What is the term for the sum of the diagonal elements of a square matrix?
- What is the trace of a square matrix whose eigenvalues are 2, 3, and 5?
- What is the trace of any skew-symmetric matrix?
- What is the unique row-equivalent matrix form that is in echelon form, has leading ones in each non-zero row, and has zeros above and below those leading ones?
- What is the value of the determinant of an orthogonal matrix?
- What name is given to a matrix that is its own inverse (A² = I)?
- What name is given to a square matrix where all entries below the main diagonal are zero?
- What property is satisfied when the inner product of two non-zero vectors is zero?
- What property must a real square matrix have to be guaranteed an orthonormal basis of eigenvectors?
- What property of a square matrix is equal to the sum of its eigenvalues?
- What specific matrix form is used when a matrix cannot be diagonalized, consisting of eigenvalues on the diagonal and ones on the superdiagonal?
- What specific property defines a skew-symmetric matrix?
- What theorem states that every square matrix satisfies its own characteristic equation?
- What type of transformation is a Householder matrix typically used to perform?
- What value is calculated as the product of the eigenvalues of a square matrix?
- Which algorithm, famously used by Google to rank web pages, essentially finds the principal eigenvector of a massive normalized link matrix?
- Which algorithm uses the power iteration method on a hyperlink matrix to rank web pages?
- Which decomposition expresses a matrix as the product of a unitary matrix, a diagonal matrix, and another unitary matrix?
- Which decomposition splits a matrix into an orthogonal matrix $Q$ and an upper triangular matrix $R$?
- Which inequality states that the absolute value of the inner product of two vectors is less than or equal to the product of their norms?
- Which law states that the signs of the eigenvalues of a symmetric matrix are invariant under a change of basis by congruence?
- Which mathematician is credited with the first modern axiomatic definition of a vector space in 1888?
- Which matrix norm is defined as the square root of the sum of the absolute squares of all its elements?
- Which matrix operation corresponds to the geometric transformation of 'flipping' a matrix over its main diagonal?
- Which matrix property is defined as the ratio of the largest to the smallest singular value?
- Which polynomial, whose roots are the eigenvalues, is identical for all matrices that are 'similar' to each other?
- Which property of the characteristic polynomial ensures that a square matrix $A$ and its transpose $A^T$ have the same eigenvalues?
- Which rule provides an explicit formula for the solution of a system of linear equations using determinants?
- Which specific matrix decomposition expresses a matrix as the product of a lower triangular matrix and its transpose for positive-definite matrices?
- Which statistical technique uses the SVD or Eigen-decomposition to reduce the dimensionality of data while preserving variance?
- Which theorem provides a way to locate the eigenvalues of a complex square matrix within circles in the complex plane?
- Which theorem states that a matrix is unitarily diagonalizable if and only if it is normal?
- Which theorem states that a real symmetric matrix can be diagonalized by an orthogonal matrix?
- Who is credited with the 'Law of Inertia' regarding the signs of eigenvalues in quadratic forms?
- Who developed the theory of matrices in an 1858 memoir and had introduced the now-standard vertical-line notation for determinants in 1841?
- Who is the mathematician credited with coining the term 'matrix' in 1850?