Chapter 1.3 Flashcards
A matrix with only one column
A column vector or a column matrix.
Def: A matrix
A rectangular array of numbers, the numbers in the array are called entries in the matrix.
A matrix with only one row
A row vector or a row matrix
The size of a matrix
The m number of rows and n number of columns denoted as mxn
Numerical quantities when discussing matrices
Scalars
A matrix with n rows and n columns
A square matrix of order n
Th entries a11, a22, … , ann of a square matrix A
The main diagonal of A
Def: two equal matrices
Two matrices with the same size and whos corresponding entries are equal
Def: the sum of two matrices of the same size
The matrix obtained by adding the entries of B to the corresponding entries of A. Matrices of different sized cannot be addes.
Def: the difference of two matrices of the same size
The matrix obtained by subtracting the entries of B from the corresponding entries of A. Matrices of different sized cannot be subtracted.
Def: A scalar multiple of A
The product cA where c is any scalar, cA is a matrix obtained by multiplying each entry of A by c.
Matrix multiplication
If A is an mxr matrix and B is an rxn matrix the product AB is the mxn matrix whos entries are determined as follows:
To find the entry in row i and column j of AB, single out row i from A and column j from B. Multiply the corresponding entries from the row and column together. The sum of the resulting products is the entry ABij.
What is required by the definition of matrix multiplication?
That the number of columns in the first factor equals the number of rows in the second factor.
If A1, A2, … , Ar are matrices of the same size and c1, c2, … , cr are scalars, then the expression of the form c1A1 + c2A2 + … + crAr is called…
A linear combination of A1, A2, … , Ar with coefficients c1, c2, … , cr
If A is an mxn matrix, and if x is an nx1 column vector, then the product Ax can be expressed as
A linear combination of the column vectors of A in which the coefficients are the entries of x
The transpose of an mxn matrix A
The definition is the nxm matix that results from interchanging the rows and columns of A, that is: the first row of A is the first column of the transpose of A
The trace of A (denoted tr(A))
If A is a square matrix, the trace is the sum of the entries along the main diagonal.
A smaller matrix achieved by inserting horisontal and vertical rules between selected rows and columns (and the name of the method)
Submatrix achieved through partitioning
The row or column method to compute rows or columns of AB
j:th column vector of AB = A[j:th column vector of B]
I:th row vector of AB = [i:th row vector of A]B
