Problem 2 - Arbeitsbuch Mathematik für Ingenieure

Question

Man zeige, dass die Zeilen einer nichtsingulären Matrix \(A \in \mathbb{R}^{n, n}\) so permutiert werden können, dass die Diagonalelemente der neu entstandenen Matrix \(\tilde{A} \in \mathbb{R}^{n, n}\) alle von Null verschieden sind.

Step-by-Step Solution

Answer

Rows of a non-singular matrix can be permuted to make all diagonal elements non-zero through Gaussian elimination.

Step 1 - Understand Non-Singular Matrices

A non-singular (invertible) matrix is one that has a non-zero determinant. This means that its rows are linearly independent.

Step 2 - Permutation of Rows

A permutation of the rows of a matrix is a reordering of the rows. Permuting the rows of a matrix does not affect the linear independence of the rows.

Step 3 - Existence of a Permutation

Since Matrix A is non-singular, each step of Gaussian elimination can always find a pivot element that is non-zero. Thus, through row permutations, it is possible to place a non-zero element on each diagonal position.

Step 4 - Constructing \(\tilde{A}\)

By appropriately permuting the rows of A, we can ensure that the resulting matrix \(\tilde{A}\) has all non-zero diagonal elements.

Step 5 - Conclusion

Therefore, the rows of a non-singular matrix A can indeed be permuted so that the diagonal elements of the resulting matrix \(\tilde{A}\) are all non-zero.

Key Concepts

Linear Independence

In linear algebra, the concept of linear independence is fundamental. Simply put, a set of vectors is linearly independent if no vector in the set can be written as a combination of the others. This means that every vector adds new information and can't be removed without losing some of the span of the set. For matrices, the rows (or columns) are considered linearly independent if the matrix is non-singular. This is directly connected to the determinant being non-zero.

Row Permutations

Row permutations involve changing the order of the rows in a matrix. This concept is key in various matrix operations, especially during Gaussian elimination. By permuting rows, you can move non-zero elements to desired positions, such as the diagonals. The big advantage here is that row permutations don't affect the linear independence of the rows. Thus, for a non-singular matrix, the matrix remains non-singular, even after row permutations.

Gaussian Elimination

Gaussian elimination is a method used to solve systems of linear equations, to find the rank of a matrix, and to determine the inverse of an invertible matrix. This method involves three types of row operations: swapping two rows, multiplying a row by a non-zero scalar, and adding or subtracting the multiple of one row to another. These operations simplify the matrix step-by-step, eventually converting it into an upper triangular form from which solutions can be easily extracted. Essential to this process is the idea of pivot elements, which are the non-zero elements used to eliminate other entries in their column.

Matrix Determinant

The determinant is a special number that can be calculated from a square matrix. It provides a lot of important information about the matrix. For instance, a matrix is invertible (non-singular) if and only if its determinant is non-zero. The determinant also reflects the volume scaling factor of the linear transformation described by the matrix. In simpler terms, if applying the matrix transformation to a region in space doesn't collapse the region to a lower dimension, the determinant will be non-zero. Conversely, a zero determinant indicates that the transformation squashes the region into a lower dimension, implying linear dependence among rows or columns.