Matrix Algebra and Invertibility: every key term you need (+ practice quiz)
25 flashcard terms for Linear Algebra Topic 2, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
A rectangular array of scalars arranged in rows and columns; a matrix with m rows and n columns is said to have size m by n and represents a map from n space to m space.
Matrix addition
The entrywise sum of two matrices of the same size. It is commutative and associative, and the zero matrix acts as the additive identity.
Scalar multiple of a matrix
The matrix obtained by multiplying every entry by a fixed scalar. Scalar multiplication distributes over both matrix addition and scalar addition.
Matrix product
For A of size m by n and B of size n by p, the entry in row i and column j of AB is the sum of products of row i of A against column j of B.
Column rule for the product
The jth column of AB equals A applied to the jth column of B, so matrix multiplication acts one column at a time on the right hand factor.
Noncommutativity of matrix products
In general AB and BA differ, may have different sizes, and one may be defined while the other is not; equality of the two products is a special situation.
Identity matrix
The square matrix with ones on the main diagonal and zeros elsewhere. It satisfies A I = A and I A = A whenever the sizes allow the products.
Zero divisors among matrices
Nonzero matrices whose product is the zero matrix. Their existence shows that cancellation of a common factor is not valid for general matrices.
Transpose
The matrix obtained by interchanging rows and columns, so the entry in row i and column j moves to row j and column i. The transpose of a product reverses the order of factors.
Symmetric matrix
A square matrix equal to its own transpose. Its entries are mirrored across the main diagonal, and such matrices behave especially well under diagonalization.
Skew symmetric matrix
A square matrix whose transpose is its negative. Every diagonal entry must therefore be zero, since each equals its own negative.
Invertible matrix
A square matrix A for which some square matrix C satisfies both A C = I and C A = I. The matrix C is then unique and is called the inverse of A.
Singular matrix
A square matrix with no inverse. Equivalently its columns are dependent, its determinant vanishes, and the homogeneous system it defines has nonzero solutions.
Inverse of a product
If A and B are invertible of the same size, then AB is invertible and its inverse is the inverse of B multiplied on the left of the inverse of A.
Two by two inverse formula
A two by two matrix with entries a, b, c, d is invertible exactly when ad minus bc is nonzero, and the inverse swaps a with d, negates b and c, and divides by that quantity.
Elementary matrix
The result of applying a single elementary row operation to an identity matrix. Multiplying on the left by such a matrix performs that same operation.
Row reduction algorithm for the inverse
Place A beside an identity matrix and row reduce. If A reduces to the identity, the block that began as the identity becomes the inverse of A.
Invertible Matrix Theorem
For a square matrix, invertibility, having a pivot in every column and row, columns forming a basis, injectivity, surjectivity, and a nonzero determinant are all equivalent statements.
One sided inverse for square matrices
For square matrices a left inverse is automatically a right inverse and conversely, so checking one product against the identity suffices.
Block matrix
A matrix partitioned into submatrices by horizontal and vertical cuts. Compatible partitions may be multiplied by treating the blocks as if they were entries.
Diagonal matrix
A square matrix whose entries away from the main diagonal are zero. Products, powers, and inverses of such matrices act separately on each diagonal entry.
Triangular matrix
A square matrix whose entries below the diagonal are all zero, or whose entries above the diagonal are all zero. It is invertible exactly when no diagonal entry is zero.
LU factorization
A writing of a matrix as a lower triangular factor times an upper triangular factor. It lets many systems with the same coefficients be solved cheaply by two triangular sweeps.
Matrix power
Repeated multiplication of a square matrix by itself. Powers describe iterated processes, and their behaviour is governed by the eigenvalues of the matrix.
Trace of a matrix
The sum of the diagonal entries of a square matrix. It satisfies the identity that the trace of AB equals the trace of BA whenever both products are defined.