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Linear Algebra
47 flashcards
A vector is a quantity that has both magnitude and direction. It can be represented geometrically as an arrow in space.
A scalar is a quantity that has only magnitude, no direction. Examples include mass, time, and temperature.
A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns.
The dimension of a vector is the number of components or entries it has.
The order of a matrix is the number of rows and columns it has, usually written as m x n, where m is the number of rows and n is the number of columns.
A linear transformation is a function that maps vectors in one vector space to vectors in another vector space while preserving vector addition and scalar multiplication.
A vector space is a set of vectors that satisfies certain algebraic operations, such as vector addition and scalar multiplication, and follows specific axioms.
The null space of a matrix is the set of all vectors x such that Ax = 0, where A is the matrix.
The rank of a matrix is the dimension of the column space or the number of linearly independent columns in the matrix.
The determinant of a matrix is a scalar value that is a function of the entries of the matrix. It is used to determine if the matrix is invertible and to solve systems of linear equations.
The transpose of a matrix is a new matrix obtained by interchanging the rows and columns of the original matrix.
The dot product of two vectors is a scalar value obtained by multiplying the corresponding components of the vectors and then summing the products.
The cross product of two vectors is a vector that is perpendicular to both vectors and has a magnitude equal to the area of the parallelogram formed by the two vectors.
The Euclidean norm of a vector is the square root of the sum of the squares of its components. It represents the length or magnitude of the vector.
The inverse of a matrix A is a matrix B such that AB = BA = I, where I is the identity matrix.
A basis for a vector space is a set of linearly independent vectors that can be used to express any vector in the space as a linear combination of the basis vectors.
The span of a set of vectors is the set of all possible linear combinations of those vectors.
The Gram-Schmidt process is a method for finding an orthonormal basis for a vector space, given a set of linearly independent vectors.
A diagonalizable matrix is a square matrix that is similar to a diagonal matrix, meaning it can be written as P^(-1)DP, where D is a diagonal matrix and P is an invertible matrix.
The trace of a square matrix is the sum of its diagonal elements.
A positive definite matrix is a symmetric matrix whose quadratic form is positive for all non-zero vectors.
The singular value decomposition of a matrix is a factorization of the matrix into the product of three matrices: U, ฮฃ, and V^T, where U and V are orthogonal matrices, and ฮฃ is a diagonal matrix.
The LU decomposition is a matrix decomposition that writes a matrix as the product of a lower triangular matrix (L) and an upper triangular matrix (U).
An eigenvalue of a square matrix is a scalar value ฮป such that Av = ฮปv for some non-zero vector v, where v is called the eigenvector corresponding to the eigenvalue ฮป.
An orthogonal matrix is a square matrix whose rows and columns are orthonormal vectors, meaning they are mutually perpendicular and have unit length.
A unitary matrix is a square complex matrix whose inverse is equal to its conjugate transpose.
A rotation matrix is a matrix that rotates a vector or a coordinate system about a fixed origin by a certain angle in a specific plane.
A projection matrix is a matrix that projects a vector onto a subspace of the vector space.
A normal matrix is a square matrix that commutes with its conjugate transpose, meaning AA^H = A^HA, where A^H is the conjugate transpose of A.
A Hermitian matrix is a square matrix that is equal to its conjugate transpose, meaning A = A^H.
A skew-symmetric matrix is a square matrix whose transpose is equal to its negative, meaning A^T = -A.
A Markov matrix is a square matrix that describes the transitions of a Markov chain, with each element representing the probability of transitioning from one state to another.
A stochastic matrix is a square matrix with non-negative entries, and each row sums to 1. It is often used to represent transition probabilities in Markov chains.
A nilpotent matrix is a square matrix whose powers eventually become the zero matrix after a finite number of multiplications.
An idempotent matrix is a square matrix that, when multiplied by itself, gives itself, meaning A^2 = A.
A permutation matrix is a square matrix that represents a permutation of a sequence or a matrix, having exactly one entry of 1 in each row and column, with the remaining entries being 0.
A Toeplitz matrix is a matrix in which each descending diagonal from left to right is constant, meaning the entries are constant along each diagonal.
A Vandermonde matrix is a matrix with the terms of a geometric progression in each row, often used in polynomial interpolation.
A Householder matrix is a matrix used in the Householder transformation, which reflects a vector across a plane or hyperplane containing the origin.
The Kronecker product, denoted by โ, is an operation on two matrices that produces a larger matrix by taking the product of each element of the first matrix with the second matrix.
A block matrix is a matrix composed of smaller matrices, or blocks, arranged in a rectangular layout.
A circulant matrix is a square matrix in which each row is a cyclic shift of the row above it, and the columns are also cyclic shifts of the first column.
A Sylvester matrix is a matrix that arises in the study of resultants and is used to solve systems of polynomial equations.
A Hankel matrix is a square matrix in which each ascending skew diagonal from left to right is constant, meaning the entries are constant along each skew diagonal.
A Cauchy matrix is a matrix whose entries are given by the reciprocal of the difference between the row and column indices, often used in the study of integral equations.
A Hadamard matrix is a square matrix whose entries are either +1 or -1, and whose rows and columns are mutually orthogonal.
The Khatri-Rao product is a column-wise Kronecker product of two matrices, often used in signal processing and multilinear algebra.