Find the values of c for which the matrix S = [[c 1 1], [1,c,1], [1,1,c]] is positive definite. Show drafts

To determine the values of \( c \) for which the matrix \( S = \begin{bmatrix} c & 1 & 1 \\ 1 & c & 1 \\ 1 & 1 & c \end{bmatrix} \) is positive definite, we need to check the following conditions:

1. All the leading principal minors of the matrix must be positive.
2. The matrix must be symmetric, which it is in this case.

### Step 1: Calculate the leading principal minors

For the matrix \( S \), the leading principal minors are:

1. **First Minor:** The determinant of the \( 1 \times 1 \) matrix formed by the first element.
2. **Second Minor:** The determinant of the \( 2 \times 2 \) leading principal submatrix.
3. **Third Minor:** The determinant of the full \( 3 \times 3 \) matrix.

Let's calculate each one.

#### First Minor:
The first minor is simply the element \( S_{11} = c \).

So, the first condition is:
\[
c > 0
\]

#### Second Minor:
The second minor is the determinant of the \( 2 \times 2 \) matrix:
\[
\text{det}\left(\begin{bmatrix} c & 1 \\ 1 & c \end{bmatrix}\right) = c^2 - 1
\]
So, the second condition is:
\[
c^2 - 1 > 0 \quad \text{or} \quad c > 1 \quad \text{or} \quad c < -1
\]

#### Third Minor:
The third minor is the determinant of the full \( 3 \times 3 \) matrix:
\[
\text{det}(S) = \text{det}\left(\begin{bmatrix} c & 1 & 1 \\ 1 & c & 1 \\ 1 & 1 & c \end{bmatrix}\right)
\]
Expanding this determinant, we get:
\[
\text{det}(S) = c(c^2 - 1 - 1) - 1(c - 1) + 1(c - 1) = c(c^2 - 2) - (c - 1) + (c - 1)
\]
Simplifying:
\[
\text{det}(S) = c^3 - 2c - (c - 1) + (c - 1) = c^3 - 2c
\]
So, the third condition is:
\[
c^3 - 2c > 0
\]
Factor this expression:
\[
c(c^2 - 2) > 0
\]

### Step 2: Analyze the conditions

To satisfy all three conditions:

1. \( c > 0 \)
2. \( c > 1 \) or \( c < -1 \)
3. \( c(c^2 - 2) > 0 \)

### Combining the conditions:

- From the second minor, \( c > 1 \) or \( c < -1 \).
- The third condition \( c(c^2 - 2) > 0 \) implies \( c > \sqrt{2} \) or \( c < -\sqrt{2} \).

The conditions are met when:

\[
c > \sqrt{2}
\]

### Conclusion:
The matrix \( S \) is positive definite for \( c > \sqrt{2} \).

Would you like any details, or have any questions?

Here are five related questions to explore further:

1. How do you determine if a matrix is positive semi-definite?
2. Can the determinant of a matrix alone determine if it's positive definite?
3. What are the eigenvalues of the matrix \( S \) for different values of \( c \)?
4. How does the concept of positive definiteness apply to quadratic forms?
5. Can a non-symmetric matrix be positive definite?

**Tip:** Always check the leading principal minors of a matrix when determining if it's positive definite.

Explore how to determine the values of c for which the matrix S = [c 1 1, 1 c 1, 1 1 c] is positive definite using leading principal minors and Sylvester's criterion. Detailed step-by-step analysis and conditions provided.

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