Find the singular values of the matrix.



To find the singular values of the matrix, we'll follow these steps:

### 1. **Given Matrix**:
\[
\mathbf{X} = \begin{pmatrix}
27 & 32 & 27 & 32 & 12 & 32 \\
13 & 43 & 58 & 28 & 53 & 33
\end{pmatrix}
\]

### 2. **Singular Value Decomposition (SVD)**:
The singular values of a matrix are the square roots of the eigenvalues of the matrix \(\mathbf{X}^\top \mathbf{X}\) (or \(\mathbf{X} \mathbf{X}^\top\)).

### 3. **Compute \(\mathbf{X}^\top \mathbf{X}\)**:
First, compute the matrix \(\mathbf{X}^\top \mathbf{X}\), where \(\mathbf{X}^\top\) is the transpose of \(\mathbf{X}\).

\[
\mathbf{X}^\top = \begin{pmatrix}
27 & 13 \\
32 & 43 \\
27 & 58 \\
32 & 28 \\
12 & 53 \\
32 & 33
\end{pmatrix}
\]

Now, compute the product:
\[
\mathbf{X}^\top \mathbf{X} = \begin{pmatrix}
27 & 13 \\
32 & 43 \\
27 & 58 \\
32 & 28 \\
12 & 53 \\
32 & 33
\end{pmatrix}
\begin{pmatrix}
27 & 32 & 27 & 32 & 12 & 32 \\
13 & 43 & 58 & 28 & 53 & 33
\end{pmatrix}
\]

Calculating each element:
\[
\mathbf{X}^\top \mathbf{X} = \begin{pmatrix}
27^2 + 13^2 & 27 \times 32 + 13 \times 43 & \ldots \\
32 \times 27 + 43 \times 13 & 32^2 + 43^2 & \ldots \\
\vdots & \vdots & \ddots
\end{pmatrix}
\]

Let me perform these calculations to find the exact matrix \(\mathbf{X}^\top \mathbf{X}\).The matrix \(\mathbf{X}^\top \mathbf{X}\) is:

\[
\mathbf{X}^\top \mathbf{X} = \begin{pmatrix}
898 & 1423 & 1483 & 1228 & 1013 & 1293 \\
1423 & 2873 & 3358 & 2228 & 2663 & 2443 \\
1483 & 3358 & 4093 & 2488 & 3398 & 2778 \\
1228 & 2228 & 2488 & 1808 & 1868 & 1948 \\
1013 & 2663 & 3398 & 1868 & 2953 & 2133 \\
1293 & 2443 & 2778 & 1948 & 2133 & 2113
\end{pmatrix}
\]

### 4. **Compute the Eigenvalues**:
The singular values of the matrix \(\mathbf{X}\) are the square roots of the eigenvalues of the matrix \(\mathbf{X}^\top \mathbf{X}\).

Let's compute these eigenvalues and then find their square roots.The singular values of the matrix \(\mathbf{X}\) are approximately:

\[
\begin{pmatrix}
117.64, & 29.998, & 0, & 0, & 0, & 0
\end{pmatrix}
\]

The other values are very close to zero and are due to numerical precision limitations.

### Explanation:
- The singular values are the non-negative square roots of the eigenvalues of \(\mathbf{X}^\top \mathbf{X}\).
- The singular values indicate the magnitude of the principal components of the matrix.

Would you like to know more about how singular values are used or have any other questions?

Here are five related questions:
1. What is the significance of the singular values in data analysis?
2. How do singular values relate to the rank of a matrix?
3. What is the relationship between singular values and eigenvalues?
4. How can singular values be used in dimensionality reduction?
5. What does it mean if a matrix has zero singular values?

**Tip:** Singular values are essential in many applications, including image compression and principal component analysis (PCA).

Learn how to find the singular values of a matrix using Singular Value Decomposition (SVD) and eigenvalue computations. This guide includes detailed calculations and explanations. Ideal for advanced undergraduate students studying linear algebra.

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