Linear Algebra Trivia Questions
- In the process of Gaussian elimination, what is the name of the non-zero entry used to eliminate entries below it in a column?
- In the Singular Value Decomposition A = UΣV^T, what type of matrices are U and V?
- In the Singular Value Decomposition A = UΣVᵀ, the columns of which matrix are called the left singular vectors?
- Which matrix decomposition of a centered data matrix yields the principal component directions as its right singular vectors?
- Sylvester's Law of Inertia describes the properties of which type of algebraic forms, typically represented by symmetric matrices?
- The dimension of the space spanned by the rows of a matrix is always equal to the dimension of which other space?
- The solution to the system of linear differential equations x'(t) = Ax(t) involves which matrix function?
- What are the diagonal elements of a triangular matrix equal to?
- What are the only two possible eigenvalues for an idempotent matrix (where A² = A)?
- What complex-valued matrix type is the generalization of a symmetric matrix, where the matrix is equal to its conjugate transpose?
- What data science technique uses the top $k$ eigenvectors of a covariance matrix to reduce dimensionality?
- What do we call a square matrix with 1s on the main diagonal and 0s everywhere else?
- What is a square matrix called if raising it to some positive integer power results in the zero matrix?
- What is a vector space equipped with an inner product that is also a complete metric space called?
- What is the Big O time complexity of the standard schoolbook algorithm for multiplying two n x n matrices?
- What is the common name for the Moore-Penrose inverse, which generalizes the concept of an inverse to non-square matrices?
- What is the computational time complexity of standard Gaussian elimination for an n x n matrix?
- What is the determinant of a matrix whose rows are linearly dependent?
- What is the determinant of an identity matrix of any size?
- What is the determinant of an n x n skew-symmetric matrix if n is an odd integer?
- What is the determinant of an upper or lower triangular matrix?
- What is the dimension of the dual space V* of a finite-dimensional vector space V?
- What is the dimension of the space of all linear functionals on a vector space V of dimension n?
- What is the dimension of the subspace of all 3x3 symmetric matrices?
- What is the dimension of the subspace of solutions to the homogeneous system $Ax = 0$ called?
- What is the dimension of the subspace spanned by the zero vector alone?
- What is the dimension of the vector space of all 2x2 real matrices?
- What is the dimension of the vector space of all $m \times n$ matrices?
- What is the dot product of any two non-zero vectors that are orthogonal to each other?
- What is the geometric interpretation of the absolute value of the determinant of a 3x3 matrix?
- What is the kernel of a linear transformation T: V -> W?
- What is the matrix obtained by taking the transpose of a matrix and then the complex conjugate of each entry?
- What is the maximum absolute value of the eigenvalues of a matrix called?
- What is the maximum number of linearly independent rows in a matrix called?
- What is the maximum number of linearly independent vectors in a vector space called?
- What is the name for a square matrix that has zeros everywhere except for the main diagonal and the diagonals immediately above and below it?
- What is the name for a square matrix where all entries off the main diagonal are zero?
- What is the name of the decomposition that factors a positive-definite matrix into the product $L L^T$?
- What is the name of the 'nearly diagonal' form every square matrix can be reduced to over the complex numbers?
- What is the name of the polynomial defined by the determinant of (A - λI)?
- What is the name of the process used to transform a set of linearly independent vectors into an orthonormal basis?
- What is the name of a row echelon form in which each leading entry is 1 and is the only non-zero entry in its column?
- What is the name of the scalar values λ for which the equation Ax = λx has a non-zero solution x?
- What is the name of the space spanned by the columns of a matrix A?
- What is the name of the subspace containing all vectors x such that Ax = 0?
- What is the name of the theorem stating that the dimension of the kernel plus the dimension of the image equals the dimension of the domain?
- What is the name of the vector space consisting of all linear functionals on a vector space V?
- What is the rank of an n x n identity matrix?
- What is the relationship between the product of the eigenvalues of a matrix and its determinant?
- What is the result of the matrix exponential e^A when A is the zero matrix?