Linear Algebra Trivia Questions
- A Jordan block is a square matrix where the eigenvalue is on the diagonal and what value is on the superdiagonal?
- A linear transformation is called an isomorphism if it is both surjective and what?
- A matrix is strictly diagonally dominant if, for every row, the magnitude of the diagonal element is greater than what?
- A real symmetric matrix is always diagonalizable by which specific type of matrix?
- A real symmetric matrix is positive definite if all of its eigenvalues satisfy what condition?
- A square matrix A is called what if its transpose is also its inverse (Aᵀ = A⁻¹)?
- A square matrix A is invertible if and only if its determinant is what value?
- A square matrix is called what if its conjugate transpose is equal to itself?
- A square matrix is called what if its transpose is equal to its negative?
- A square matrix that is equal to its own transpose is called what?
- Which numerical application uses a Vandermonde matrix to determine the coefficients of a polynomial passing through a given set of data points?
- According to Sylvester's Law of Inertia, what three counts remain invariant for a given real quadratic form?
- According to the Rank-Nullity Theorem, if a matrix has 7 columns and a rank of 3, what is its nullity?
- According to the Rank-Nullity Theorem, what is the sum of the rank and the nullity of an m x n matrix?
- Eigenvectors corresponding to distinct eigenvalues of a real symmetric matrix are always what?
- For a 2x2 matrix with entries a, b (top row) and c, d (bottom row), what is the determinant formula?
- For a real matrix A, the non-zero singular values are the square roots of the non-zero eigenvalues of which two matrices?
- For a real symmetric matrix, what property is guaranteed for all its eigenvalues?
- For a square matrix A, what is the power series definition of the matrix exponential, e^A?
- For an m x n matrix A with rank r, what is the dimension of its null space?
- If a linear transformation T: V -> W is surjective, what is the relationship between the dimension of the image and the dimension of W?
- If a matrix A is similar to a diagonal matrix D, the matrix A is said to be what?
- If a matrix A is skew-symmetric (Aᵀ = -A), what must be true about its diagonal entries?
- If a matrix P satisfies the condition P² = P, it is representing which type of linear transformation?
- If a set of n vectors in an n-dimensional space is linearly independent, what can be said about the determinant of the matrix they form?
- If a square matrix A has a determinant of zero, what name is given to this type of matrix?
- If a square matrix A is diagonalizable, it must have n linearly independent what?
- If a square matrix A is nilpotent, what is its only eigenvalue?
- If matrix A is related to matrix B by B = P⁻¹AP, what is the term for the relationship between A and B?
- If P is an orthogonal projection matrix, what is the result of the operation P times P?
- If P is the transition matrix from basis B to basis B', what is the transition matrix from B' to B?
- If the geometric multiplicity of an eigenvalue is strictly less than its algebraic multiplicity, the matrix is said to be what?
- If two rows of a square matrix are identical, what is the value of its determinant?
- In a column-stochastic matrix used for Markov chains, what must the sum of each column be?
- In a Jordan Canonical Form block, what values are located on the main diagonal?
- In a Markov chain with transition matrix P, what do we call a probability vector v that satisfies the equation vP = v?
- In a Markov chain, what do we call a square matrix used to describe the transitions of a system?
- In a Markov chain, what is the name for a matrix where every column sums to 1 and all entries are non-negative?
- In a right stochastic transition matrix of a Markov chain, what must the entries in each row sum to?
- In a system of linear equations Ax=0, if the coefficient matrix has more columns than rows (m < n), the system must have at least one of what type of variable?
- In a vector space, what is the term for a set of vectors that are linearly independent and span the entire space?
- In determinant calculation, what is the term for a signed minor of a matrix element?
- In linear algebra, what is the term for the set of all vectors v such that Av = 0?
- In mathematics, what is the term for a rectangular array of numbers, symbols, or expressions arranged in rows and columns?
- In Principal Component Analysis (PCA), the first principal component is the eigenvector associated with which eigenvalue?
- In Singular Value Decomposition (SVD), the diagonal elements of the central matrix are known by what name?
- What is a square matrix called if the real parts of all its eigenvalues are strictly negative?
- In the context of linear transformations, what is the 'kernel'?
- In the context of numerical linear algebra, what does the 'condition number' of a matrix measure?
- In the context of the Jordan Canonical Form, what are the entries located immediately above the diagonal in a Jordan block?