matriks A= [1 11 23 31. 2 13 21 33. 3 12 22 32] varians(x)=(34,60; 242,14; 41,02; 280,16). Varians(Ax)?

Given matrix \( A \) and the variance vector \( \text{Var}(x) \), we aim to find the variance of the transformed vector \( A \cdot x \).

### Matrix \( A \) is defined as:
\[
A = \begin{bmatrix}
1 & 11 & 23 & 31 \\
2 & 13 & 21 & 33 \\
3 & 12 & 22 & 32
\end{bmatrix}
\]

### Variance of \( x \) is given as:
\[
\text{Var}(x) = \begin{bmatrix} 34.60 \\ 242.14 \\ 41.02 \\ 280.16 \end{bmatrix}
\]

Now, to find \( \text{Var}(A \cdot x) \), we need to use the fact that:

\[
\text{Var}(A \cdot x) = A \cdot \text{Var}(x) \cdot A^\top
\]

This formula takes into account the linear transformation properties of variances. 

Let me calculate \( \text{Var}(A \cdot x) \).The variance of the transformed vector \( A \cdot x \) is:

\[
\text{Var}(A \cdot x) = \begin{bmatrix}
320266.88 & 341111.56 & 330741.12 \\
341111.56 & 364244.12 & 352781.64 \\
330741.12 & 352781.64 & 341917.08
\end{bmatrix}
\]

This represents the covariance matrix of the transformed data.

Do you need more details on how this calculation was done or have any questions?

Here are five related questions:
1. How do you interpret a covariance matrix?
2. What are the eigenvalues and eigenvectors of this covariance matrix?
3. How does a linear transformation affect the variance and covariance of a random vector?
4. Can this method be extended to larger matrices?
5. How can we compute the correlation matrix from this covariance matrix?

**Tip:** The diagonal entries of a covariance matrix represent the variances of individual components, while the off-diagonal entries represent the covariances between pairs of components.

The query submitted by the user: matriks A= [1 11 23 31. 2 13 21 33. 3 12 22 32] varians(x)=(34,60; 242,14; 41,02; 280,16). Varians(Ax)?

Explore how to calculate the variance of a transformed vector using matrix multiplication. Given matrix A and the variance vector Var(x), we find the covariance matrix Var(Ax) using the formula Var(Ax) = A * Var(x) * Aᵀ. This detailed solution includes step-by-step calculations and an explanation of variance properties under linear transformations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation