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Magazine Chapter 6: Problem 6
Zeigen Sie, dass jede Matrix der Form \(k \cdot E_{n}\) mit jeder Matrix aus \(K^{n \times n}\) vertauschbar ist.
Short Answer
Expert verified
Matrix \(k \cdot E_{n}\) commutes with any matrix \(A\) in \(K^{n \times n}\).
Step by step solution
01
Understanding the Initial Problem
We need to show that a matrix of the form \(k \cdot E_{n}\) commutes with any matrix \(A\) from \(K^{n \times n}\). This means proving that \(k \cdot E_{n} \times A = A \times k \cdot E_{n}\).
02
Matrix Multiplication with Scalar
The matrix \(k \cdot E_{n}\) is a scalar multiple of the identity matrix \(E_n\), where each diagonal element of the identity matrix is multiplied by \(k\). This results in a diagonal matrix where all diagonal entries are \(k\), and all off-diagonal entries are 0.
03
Calculate Left Product
To compute \(k \cdot E_{n} \times A\), we use the property of identity matrices that multiplying any matrix by the identity results in the original matrix scaled by \(k\). Thus, \((k \cdot E_{n}) \times A = k \times A\), meaning every element of \(A\) is multiplied by \(k\).
04
Calculate Right Product
Similarly, to compute \(A \times k \cdot E_{n}\), the scalar multiplication will distribute through the multiplication just like the left-hand product. Hence, \(A \times (k \cdot E_{n}) = k \times A\).
05
Equivalence Confirmation
Since both the left product \((k \cdot E_{n}) \times A\) and the right product \(A \times (k \cdot E_{n})\) yield \(k \times A\), they are equal, proving that \(k \cdot E_{n}\) commutes with any matrix \(A\) in \(K^{n \times n}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Scalar Multiplication
Scalar multiplication is a fundamental operation in matrix algebra. It involves multiplying every element of a matrix by a constant, known as the scalar. If you have a matrix \( A \) and a scalar \( k \), scalar multiplication results in a new matrix \( k \, A \), where each element of \( A \) is multiplied by \( k \).
- For example, if \( A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \) and \( k = 3 \), then \( k \cdot A = \begin{bmatrix} 3 & 6 \ 9 & 12 \end{bmatrix} \).
- This operation is straightforward and scales the matrix uniformly.
Identity Matrix
The identity matrix is a special kind of square matrix which acts like the number 1 does in regular multiplication. Denoted as \( E_n \) for an \( n \times n \) matrix, its major characteristic is that its diagonal elements are all 1, and its off-diagonal elements are all 0.
- Basic example: For a 3x3 identity matrix, we have \( E_3 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \).
- When any matrix is multiplied by the identity matrix of compatible dimensions, the result is the original matrix: \( A \times E = A \).
Matrix Algebra
Matrix algebra is a set of mathematical rules and operations for performing algebraic manipulations with matrices. At its core are operations like addition, subtraction, and multiplication, each with specific properties and requirements.
- Matrix addition involves adding corresponding elements of matrices of the same dimension.
- Matrix multiplication requires the inner dimensions of the matrices to match and involves summing the products of corresponding entries across rows and columns.
- Commutative properties do not usually apply to matrix multiplication. That is, for two matrices \( A \) and \( B \), generally, \( A \times B eq B \times A \).
Diagonal Matrix
A diagonal matrix is a type of matrix where all off-diagonal elements are zero. The diagonal elements, however, can be non-zero or zero. A common example of a diagonal matrix is the identity matrix.
- Represented as \( D = \begin{bmatrix} d_{11} & 0 & \dots & 0 \ 0 & d_{22} & \dots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \dots & d_{nn} \end{bmatrix} \).
- Diagonal matrices are simple to work with in algebra because their multiplication and inversion are straightforward.
- In this scenario, \( k \cdot E_n \) is itself a diagonal matrix where each diagonal entry is \( k \).
