What's A Pivot In Linear Algebra

In linear algebra, a pivot is a crucial element used in the process of solving systems of linear equations and performing Gaussian elimination. Understanding pivots is fundamental to grasping many concepts in matrix operations and linear transformations. This comprehensive guide will delve into the definition of a pivot, its significance, how to find it, and its applications in various areas of linear algebra.

Definition of a Pivot

A pivot, in the context of linear algebra, refers to the first non-zero element in a row of a matrix, typically encountered during Gaussian elimination or row reduction to echelon form. This element is used to eliminate the entries below it in the same column, thereby simplifying the matrix and allowing for the solution of linear equations.

More formally, a pivot satisfies the following conditions:

• It is a non-zero entry in a matrix.
• It is the first non-zero entry in its row when scanning from left to right.
• It is used as a reference point to eliminate other entries in the same column.

The location of a pivot (i.e., its row and column indices) and its value are both significant. Pivots help determine the rank of a matrix, identify linearly independent rows or columns, and solve systems of linear equations.

Importance of Pivots

Pivots play a critical role in several aspects of linear algebra:

1. Solving Linear Equations: Pivots are instrumental in transforming a system of linear equations into an easily solvable form. Through Gaussian elimination, the matrix representing the system is converted into an echelon form where the solution can be obtained via back-substitution.

2. Determining Matrix Rank: The number of pivots in a matrix is equal to its rank. The rank of a matrix indicates the number of linearly independent rows or columns and provides insights into the solution space of the associated linear equations.

3. Identifying Linear Independence: Each pivot corresponds to a linearly independent row and column. Therefore, by counting the number of pivots, one can determine the number of linearly independent vectors in a set.

4. Computing Matrix Inverses: Pivots are used in the Gauss-Jordan method to compute the inverse of a matrix. By augmenting the matrix with an identity matrix and performing row operations until the original matrix is transformed into the identity matrix, the inverse is revealed on the right side.

5. Performing LU Decomposition: Pivots are used in LU decomposition to factorize a matrix into a product of lower (L) and upper (U) triangular matrices. This decomposition is useful for solving linear systems more efficiently, especially when the coefficient matrix remains the same but the right-hand side vector changes.

How to Find Pivots

Finding pivots involves systematically reducing a matrix to its row echelon form using Gaussian elimination. The process can be broken down into the following steps:

1. Start with the First Column: Look at the first column of the matrix. Find the first non-zero entry in this column. This will be the pivot for the first row. If the first entry is zero, swap the first row with a row below it that has a non-zero entry in the first column. If the entire column is zero, move to the next column.

2. Eliminate Entries Below the Pivot: Once a pivot is found, use it to eliminate all entries below it in the same column. This is done by performing row operations of the form:

   Row i = Row i - (entry in Row i / pivot) * Pivot Row

   For example, if the pivot is in Row 1, Column 1, and you want to eliminate the entry in Row 2, Column 1, the operation would be:

   Row 2 = Row 2 - (entry in Row 2, Column 1 / pivot) * Row 1

3. Move to the Next Row and Column: After eliminating the entries below the pivot, move to the next row and the next column to the right. Repeat the process to find the next pivot.

4. Repeat Until Echelon Form is Reached: Continue this process until you reach the last row or until all remaining columns are zero. The matrix is then in row echelon form.

Example:

Consider the following matrix:

A = | 2  1  1 |
    | 4  3  3 |
    | 8  7  9 |

1. First Pivot: The first non-zero entry in the first column is 2 (in the first row). So, the first pivot is 2.

2. Eliminate Entries Below:

   • To eliminate the 4 in the second row, perform the operation:

     Row 2 = Row 2 - (4/2) * Row 1 = Row 2 - 2 * Row 1

   • To eliminate the 8 in the third row, perform the operation:

     Row 3 = Row 3 - (8/2) * Row 1 = Row 3 - 4 * Row 1

   The matrix becomes:

A = | 2  1  1 |
    | 0  1  1 |
    | 0  3  5 |

3. Second Pivot: Move to the second row and second column. The first non-zero entry is 1. So, the second pivot is 1.

4. Eliminate Entries Below:

   • To eliminate the 3 in the third row, perform the operation:

     Row 3 = Row 3 - (3/1) * Row 2 = Row 3 - 3 * Row 2

   The matrix becomes:

A = | 2  1  1 |
    | 0  1  1 |
    | 0  0  2 |

5. Third Pivot: Move to the third row and third column. The first non-zero entry is 2. So, the third pivot is 2.

The final matrix in row echelon form is:

A = | 2  1  1 |
    | 0  1  1 |
    | 0  0  2 |

The pivots are 2, 1, and 2.

Special Cases and Considerations

1. Zero Pivots: If, during the Gaussian elimination process, you encounter a zero in the position where a pivot should be, you must perform a row swap. Swap the row with a row below it that has a non-zero entry in the same column. If no such row exists, move to the next column.

2. Matrices with All Zero Rows: If a matrix has rows consisting entirely of zeros, these rows are placed at the bottom of the matrix during row reduction. These rows do not contain pivots and do not contribute to the rank of the matrix.

3. Non-Square Matrices: The same principles of finding pivots apply to non-square matrices. Gaussian elimination is performed until the matrix is in row echelon form, and the pivots are identified as the first non-zero entries in each row.

Applications of Pivots

1. Solving Systems of Linear Equations

Pivots are essential for solving systems of linear equations. The process involves converting the augmented matrix of the system into row echelon form and then using back-substitution to find the values of the variables.

Example:

Consider the system of linear equations:

2x + y + z = 8
4x + 3y + 3z = 18
8x + 7y + 9z = 50

The augmented matrix for this system is:

| 2  1  1 | 8 |
| 4  3  3 | 18|
| 8  7  9 | 50|

Performing Gaussian elimination (as shown in the previous example) results in the row echelon form:

| 2  1  1 | 8 |
| 0  1  1 | 2 |
| 0  0  2 | 8 |

Now, using back-substitution:

• From the third row: 2z = 8 => z = 4
• From the second row: y + z = 2 => y + 4 = 2 => y = -2
• From the first row: 2x + y + z = 8 => 2x - 2 + 4 = 8 => 2x = 6 => x = 3

Thus, the solution to the system of equations is x = 3, y = -2, and z = 4.

2. Determining Matrix Rank

The rank of a matrix is the number of non-zero rows in its row echelon form, which is equal to the number of pivots. The rank provides information about the dimensionality of the vector space spanned by the rows or columns of the matrix.

Example:

Consider the matrix:

A = | 1  2  3 |
    | 2  4  6 |
    | 3  6  9 |

Performing Gaussian elimination:

Row 2 = Row 2 - 2 * Row 1
Row 3 = Row 3 - 3 * Row 1

The matrix becomes:

A = | 1  2  3 |
    | 0  0  0 |
    | 0  0  0 |

There is only one pivot (1), so the rank of the matrix is 1. This indicates that only one row (or column) is linearly independent.

3. Computing Matrix Inverses

Pivots are used in the Gauss-Jordan method to compute the inverse of a matrix. The process involves augmenting the matrix with an identity matrix and performing row operations until the original matrix is transformed into the identity matrix. The inverse is then found on the right side.

Example:

Consider the matrix:

A = | 2  1 |
    | 1  1 |

Augment the matrix with the identity matrix:

| 2  1 | 1  0 |
| 1  1 | 0  1 |

Perform row operations to get the left side into the identity matrix:

1. Row 1 = Row 1 / 2

| 1  0.5 | 0.5  0 |
| 1  1   | 0    1 |

2. Row 2 = Row 2 - Row 1

| 1  0.5 | 0.5  0 |
| 0  0.5 | -0.5 1 |

3. Row 2 = Row 2 / 0.5

| 1  0.5 | 0.5  0 |
| 0  1   | -1   2 |

4. Row 1 = Row 1 - 0.5 * Row 2

| 1  0 | 1  -1 |
| 0  1 | -1  2 |

The inverse of the matrix A is:

A^-1 = | 1  -1 |
       | -1  2 |

4. LU Decomposition

LU decomposition is a method to factorize a matrix into a lower triangular matrix (L) and an upper triangular matrix (U). Pivots play a role in ensuring the stability and accuracy of this decomposition.

Example:

Consider the matrix:

A = | 2  1  1 |
    | 4  3  3 |
    | 8  7  9 |

Performing Gaussian elimination without row swaps, we already have the upper triangular matrix (U):

U = | 2  1  1 |
    | 0  1  1 |
    | 0  0  2 |

The lower triangular matrix (L) consists of the factors used in the row operations during Gaussian elimination:

L = | 1  0  0 |
    | 2  1  0 |
    | 4  3  1 |

Thus, A = LU.

LU decomposition is particularly useful for solving multiple systems of linear equations with the same coefficient matrix but different right-hand side vectors.

Conclusion

Pivots are fundamental to understanding and applying various concepts in linear algebra. They are essential for solving systems of linear equations, determining matrix rank, computing matrix inverses, and performing LU decomposition. Mastering the process of finding pivots and understanding their significance will greatly enhance your ability to work with matrices and solve linear algebra problems. Whether you are solving engineering problems, analyzing data, or working with computer graphics, a solid understanding of pivots is invaluable.