Examples of square matrix in the following topics:
-
- The determinant of a $2\times 2$ square matrix is a mathematical construct used in problem solving that is found by a special formula.
- In the case where the matrix entries are written out in full, the determinant is denoted by surrounding the matrix entries by vertical bars instead of the brackets or parentheses of the matrix.
- In linear algebra, the determinant is a value associated with a square matrix.
- For a $2 \times 2$ matrix, $\begin{bmatrix} a & b\\ c & d \end{bmatrix}$,
- Explain what a determinant represents, how to find one, and why only square matrices have them
-
- The matrix that has this property is referred to as the identity matrix.
- What matrix has this property?
- For a $3 \times 3$ matrix, the identity matrix is a $3 \times 3$ matrix with diagonal $1$s and the rest equal to $0$:
- There is no identity for a non-square matrix because of the requirement of matrices being commutative.
- For a non-square matrix $[A]
$ one might be able to find a matrix $[I]$ such that $[A][I]=[A]$, however, if the order is reversed then an illegal multiplication will be left.
-
- Augmented matrix: an augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices.
- Upper triangle form: A square matrix is called upper triangular if all the entries below the main diagonal are zero.
- A triangular matrix is one that is either lower triangular or upper triangular.
- A matrix that is both upper and lower triangular is a diagonal matrix.
- Use elementary row operations to put a matrix in simplified form
-
- The cofactor of an entry $(i,j)$ of a matrix $A$ is the signed minor of that matrix.
- In linear algebra, the cofactor (sometimes called adjunct) describes a particular construction that is useful for calculating both the determinant and inverse of square matrices.
- Specifically the cofactor of the $(i,j)$ entry of a matrix, also known as the $(i,j)$ cofactor of that matrix, is the signed minor of that entry.
- In linear algebra, a minor of a matrix $A$ is the determinant of some smaller square matrix, cut down from $A$ by removing one or more of its rows or columns.
- Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors.
-
- The matrix has a long history of application in solving linear equations.
- A matrix with m rows and n columns is called an m × n matrix or $m$-by-$n$ matrix, while m and n are called its dimensions.
- A matrix which has the same number of rows and columns is called a square matrix.
- In some contexts, such as computer algebra programs, it is useful to consider a matrix with no rows or no columns, called an empty matrix.
- Each element of a matrix is often denoted by a variable with two subscripts.
-
- Now suppose that we multiply this adjacency matrix times itself (i.e. raise the matrix to the 2nd power, or square it).
- The calculation of the matrix squared is shown as figure 5.14.
- This matrix (i.e. the adjacency matrix squared) counts the number of pathways between two nodes that are of length two.
- The adjacency matrix squared tells us how many pathways of length two are there from each actor to each other actor.
- If we took the Boolean squared matrix and multiplied it by the adjacency matrix using Boolean multiplication, the result would tell us which actors were connected by one or more pathways of length three.
-
- The matrix $B$ is the inverse of the matrix $A$ if when multiplied together, $A\cdot B$ or $B\cdot A$ gives the identity matrix.
- Note also that only square matrices can have an inverse.
- The definition of an inverse matrix is based on the identity matrix $[I]$, and it has already been established that only square matrices have an associated identity matrix.
- When multiplying this mystery matrix by our original matrix, the result is $[I]$.
- In some cases, the inverse of a square matrix does not exist.
-
- An adjacency matrix is a square actor-by-actor (i=j) matrix where the presence of pair wise ties are recorded as elements.
- The main diagonal, or "self-tie" of an adjacency matrix is often ignored in network analysis.
- Sociograms, or graphs of networks can be represented in matrix form, and mathematical operations can then be performed to summarize the information in the graph.
- Such data are represented as a series of matrices of the same dimension with the actors in the same position in each matrix.
- Many of the same tools that we can use for working with a single matrix (matrix addition and correlation, blocking, etc.)
-
- A "3 by 6" matrix has three rows and six columns; an "I by j" matrix has I rows and j columns.
- Figure 5.1 shows a two-by-four matrix.
- The cell addresses have been entered as matrix elements in the two examples above.Matrices are often represented as arrays of elements surrounded by vertical lines at their left and right, or square brackets at the left and right.
- The matrix in figure 5.3 for example, is a 4 by 4 matrix, with additional labels.
- The matrices used in social network analysis are frequently "square. " That is, they contain the same number of rows and columns.
-
- A matrix is "3 by 2. " How many columns does it have?
- There is a "1" in cell 3,2 of an adjacency matrix representing a sociogram.
- What does it mean to "permute" a matrix, and to "block" it?
- Try permuting your matrix, and blocking it.
- Can you make an adjacency matrix to represent the "star" network?