Matrix definitions Flashcards
What is the definition of the Rank of a matrix?
The Rank of a Matrix You can think of an r x c matrix as a set of r row vectors, each having c elements; or you can think of it as a set of c column vectors, each having r elements. The rank of a matrix is defined as (a) the maximum number of linearly independent column vectors in the matrix or (b) the maximum number of linearly independent row vectors in the matrix. Both definitions are equivalent. For an r x c matrix, If r is less than c, then the maximum rank of the matrix is r. If r is greater than c, then the maximum rank of the matrix is c. The rank of a matrix would be zero only if the matrix had no elements. If a matrix had even one element, its minimum rank would be one.
What is a Symmetric Matrix
Symmetric matrix. If the transpose of a matrix is equal to itself, that matrix is said to be symmetric. Two examples of symmetric matrices appear below. A = A’ = 1 2 2 3 B = B’ = 5 6 7 6 3 2 7 2 1 Note that each of these matrices satisfy the defining requirement of a symmetric matrix: A = A’ and B = B’.
What does ‘rref’ stand for?
Row reduced echelon form
What are the elementary row operations that are allowed when working with matrices?
Elementary row operations An elementary row operation is any one of the following moves: Swap: Swap two rows of a matrix. Scale: Multiply a row of a matrix by a nonzero constant. Pivot: Add a multiple of one row of a matrix to another row. Two matrices A and B are row equivalent if it is possible to transform A into B by a sequence of elementary row operations.
What are the two characteristics of a matrix in echelon form?
- Any rows that contain all zeros are found at the bottom of the matrix.
- 2: The first nonzero entry on each row is found to the right of the first nonzero entry in the preceding row. 3: ( A third possible requirement found in some texts is that the leading coefficient in each row must be 1)
What is another name for ‘Canonical Form’?
Standard Form
What is another name for the Canonical Form of a matrix?
Reduced Row Echelon Form
What is a matrix’s column space
In linear algebra, the column space (also called the range or image) of a matrix A is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matrix transformation.
What is the definition of a matrix Column Space
In linear algebra, the column space (also called the range or image) of a matrix A is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matrix transformation.
What are Elementary Matrices?
In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation.
Each type of elementary operation may be performed by matrix multiplication, using square matrices called elementary operators
How so you perform elementary row operations using elementary matrices?
How to Perform Elementary Row Operations
To perform an elementary row operation on a A, an r x c matrix, take the following steps.
To find E, the elementary row operator, apply the operation to an r x r identity matrix.
To carry out the elementary row operation, premultiply A by E.
We illustrate this process below for each of the three types of elementary row operations.
What operations do you perform to exchange two rows of a matrix using elementary matrices?
Interchange two rows. Suppose we want to interchange the second and third rows of A, a 3 x 2 matrix. To create the elementary row operator E, we interchange the second and third rows of the identity matrix I3.
Then, to interchange the second and third rows of A, we premultiply A by E, as shown below.
What does an elementary matrix that swaps the 2nd and 2rd row of a 3X3 matrix look like?
What is a traspose matrix?
A transpose is a matrix where the rows of matrix A are the columns of matrix AT
