Essential Vocabulary
Here is a link to a list of these terms on Quizlet
Column matrix (or column vector)
A matrix with rows and only 1 column.
Diagonal matrix
A matrix where whenever
Dimensions of a matrix
An matrix is a matrix with rows and columns.
Elementary matrix
A matrix formed by applying one elementary row operation to the identity matrix.
Homogeneous system
A system of linear equations is called homogeneous if the system can be written in the form or as a matrix equation as .
Identity matrix
A square matrix with ones as diagonal entries and zeros for the remaining entries.
Inverse of a square matrix
Let be an matrix. An matrix is called an inverse of if where is an identity matrix. If such an inverse matrix exists, we say that is invertible. If an inverse does not exist, we say that is not invertible. The inverse of is denoted by .
LU factorization
A factorization where is lower triangular and is upper triangular with ones on the diagonal (called unit upper triangluar). It is useful for solving .
Matrix
A rectangular array of numbers. It has rows and columns for some positive integers and .
Matrix addition
Let and be two matrices. Then the sum of matrices and , denoted by , is an matrix given by
Matrix factorization
Representing a matrix as a product of two or more matrices.
Matrix multiplication
Let be an matrix whose rows are vectors , . Let be an matrix with columns . Then the matrix product is an matrix with entries given by the dot products
Matrix powers
If is a square matrix then we can define to be the result of multiplying by itself times.
Negative of a matrix
The additive inverse of a matrix, formed by multiplying the matrix by the scalar .
Nonsingular matrix
A square matrix is said to be nonsingular provided that . Otherwise we say that is singular.
Partitioned matrices - block multiplication
Subdividing a matrix into submatrices using imaginary horizontal and vertical lines - used to multiply matrices more efficiently.
Permutation matrix
A matrix formed by permuting the rows of the identity matrix.
Properties of matrix algebra
Addition and scalar multiplication of matrices have many nice properties. Matrix multiplication is not commutative, but the associative and distributive laws hold. The inverse and transpose operations of a product follow the ”shoes and socks” rule.
Row matrix
A matrix with only row and columns.
Scalar multiple of a matrix
If and is a scalar, then .
Singular matrix
A square matrix is said to be singular provided that is NOT the identity matrix. If instead , we say that is nonsingular.
Square matrix
A matrix with the same number of rows and columns.
Symmetric matrix
An matrix is said to be symmetric if It is said to be skew symmetric if
Transpose of a matrix
Let be an matrix. Then the transpose of , denoted by , is the matrix given by switching the rows and columns:
Zero matrix
The zero matrix is the matrix having every entry equal to zero. The zero matrix is denoted by .
