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Chapter 16: Problem 5

Ist das Produkt symmetrischer Matrizen stets wieder eine symmetrische Matrix?

Short Answer

Expert verified
Answer: No, the product of two symmetric matrices is not always a symmetric matrix.

Step by step solution

01

Define a symmetric matrix

A matrix A is symmetric if its transpose is equal to itself, i.e., A^T = A.
02

Given two symmetric matrices

Suppose we have two symmetric matrices A and B, such that: \(A^T = A\) and \(B^T = B\)
03

Find the product of A and B

We need to find the product C of these matrices A and B, such that: \(C = AB\)
04

Check if the transpose of the product is equal to the product

In order to check if C is symmetric, we need to see if its transpose is equal to itself, i.e., \(C^T = C\). We can start by finding the transpose of C.
05

Find the transpose of C

We know that \(C = AB\), so to find the transpose of C, we have: \(C^T = (AB)^T\)
06

Using the transpose of a product rule

We can use the rule \((AB)^T = B^TA^T\), where \((AB)^T\) is the transpose of the product of A and B, so: \(C^T = (AB)^T = B^TA^T\)
07

Substitute the given condition of symmetric matrices

We know that \(A^T = A\) and \(B^T = B\), so we can substitute these values into the equation: \(C^T = B^TA^T = BA\)
08

Conclusion

We found that \(C^T = BA\), but to determine if \(C\) is symmetric, we need to check if \(C^T = C\). In this case, since \(C^T = BA\), and \(C = AB\), we see that: \(C^T \neq C\) As a result, the product of two symmetric matrices is not always a symmetric matrix.

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Most popular questions from this chapter

Hat eine Matrix \(\boldsymbol{A} \in \mathbb{R}^{n \times n}\) mit \(n \in 2 \mathbb{N}+1\) und \(A=-A^{\mathrm{T}}\) die Determinante \(0 ?\)

Matrix der Form $$ M=\left(\begin{array}{ll} A & C \\ 0 & B \end{array}\right) \in \mathbb{K}^{n \times n} $$ wobei \(\mathbf{0} \in \mathbb{K}^{(n-m) \times m}\) die Nullmatrix ist und \(\boldsymbol{A} \in \mathbb{K}^{m \times m}, \boldsymbol{C} \in\) \(\mathbb{K}^{m \times(n-m)}, \boldsymbol{B} \in \mathbb{K}^{(n-m) \times(n-m)}\) sind. Zeigen Sie: \(\operatorname{det} \boldsymbol{M}=\operatorname{det} \boldsymbol{A} \operatorname{det} \boldsymbol{B}\).

Ist das Produkt quadratischer oberer bzw. unterer Dreiecksmatrizen wieder eine obere bzw. untere Dreiecksmatrix?

Bestimmen Sie ein lineares Gleichungssystem. dessen L枚sungsmenge \(L=\left\langle\left(\begin{array}{c}1 \\ 0 \\\ -1\end{array}\right),\left(\begin{array}{c}0 \\ 1 \\\ -1\end{array}\right)\right\rangle \subseteq \mathbb{R}^{3}\) ist.

Gilt f眉r invertierbare Matrizen \(A, B \in \mathbb{K}^{n \times n}\) $$ \operatorname{ad}(\boldsymbol{A} \boldsymbol{B})=\operatorname{ad}(\boldsymbol{B}) \operatorname{ad}(\boldsymbol{A}) ? $$
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