Consider the $p \times p$ sample covariance matrix:

$$\mathbf{S} = \frac{1}{n-1} \cdot \mathbf{Y}_\mathbf{c}^\text{T} \mathbf{Y}_\mathbf{c} \quad \quad \quad \mathbf{Y}_\mathbf{c} = \mathbf{C} \mathbf{Y},$$

where $\mathbf{C} = \mathbf{I}-\frac{1}{n} \mathbf{1} \mathbf{1}^\text{T}$ is the $n \times n$ centering matrix and $\mathbf{Y}$ is an $n \times p$ matrix. How can it be proved that if $n-1> p$ then the sample covariance matrix is ​​positive definite?

The following clue is found in Ranchera's book:

Consider the $p \times p$ sample covariance matrix:

$$\mathbf{S} = \frac{1}{n-1} \cdot \mathbf{Y}_\mathbf{c}^\text{T} \mathbf{Y}_\mathbf{c} \quad \quad \quad \mathbf{Y}_\mathbf{c} = \mathbf{C} \mathbf{Y},$$

where $\mathbf{C} = \mathbf{I}-\frac{1}{n} \mathbf{1} \mathbf{1}^\text{T}$ is the $n \times n$ centering matrix and $\mathbf{Y}$ is an $n \times p$ matrix. How can it be proved that if the variables are continuos, not linearly related and $n-1> p$ then the sample covariance matrix is ​​positive definite?

The following clue is found in Ranchera's book:

Consider the $p \times p$ sample covariance matrix:

$$\mathbf{S} = \frac{1}{n-1} \cdot \mathbf{Y}_\mathbf{c}^\text{T} \mathbf{Y}_\mathbf{c} \quad \quad \quad \mathbf{Y}_\mathbf{c} = \mathbf{C} \mathbf{Y},$$

where $\mathbf{C} = \mathbf{I}-\frac{1}{n} \mathbf{1} \mathbf{1}^\text{T}$ is the $n \times n$ centering matrix and $\mathbf{Y}$ is an $n \times p$ matrix. How can it be proved that if $n-1> p$ then the sample covariance matrix is ​​positive definite?

Consider the $p \times p$ sample covariance matrix:

$$\mathbf{S} = \frac{1}{n-1} \cdot \mathbf{Y}_\mathbf{c}^\text{T} \mathbf{Y}_\mathbf{c} \quad \quad \quad \mathbf{Y}_\mathbf{c} = \mathbf{C} \mathbf{Y},$$

where $\mathbf{C} = \mathbf{I}-\frac{1}{n} \mathbf{1} \mathbf{1}^\text{T}$ is the $n \times n$ centering matrix and $\mathbf{Y}$ is an $n \times p$ matrix. How can it be proved that if $n-1> p$ then the sample covariance matrix is ​​positive definite?

The following clue is found in Ranchera's book:

Let $S=\frac{1}{n-1}Y_c^TY_c$ be a sample covariance matrix $nxn$, where $Y_c=(I-\frac{1}{n}J)Y$ is a matrix. $(I-\frac{1}{n}J)$ is the centering matrix and $Y$ is $nxp$ matrix. $n$: random sample observations and $p$: components of the vector: $x_i$.

How can it be proved that if $n-1> p$ then the matrix is ​​positive definite?

Consider the $p \times p$ sample covariance matrix:

$$\mathbf{S} = \frac{1}{n-1} \cdot \mathbf{Y}_\mathbf{c}^\text{T} \mathbf{Y}_\mathbf{c} \quad \quad \quad \mathbf{Y}_\mathbf{c} = \mathbf{C} \mathbf{Y},$$

where $\mathbf{C} = \mathbf{I}-\frac{1}{n} \mathbf{1} \mathbf{1}^\text{T}$ is the $n \times n$ centering matrix and $\mathbf{Y}$ is an $n \times p$ matrix. How can it be proved that if $n-1> p$ then the sample covariance matrix is ​​positive definite?