Let \Sigma denote a covariance matrix for some random vector \mathrm{{\boldsymbol X}} \in \mathbb {R}^ d. (Assume that \mathbf E[\left\| \mathrm{{\boldsymbol X}} \right\| _2^2] < \infty.)
Which of the following properties does \Sigma necessarily have? (Choose all that apply.)
Symmetric
Positive Semidefinite
Positive Definite
Orthogonal
The properties that the covariance matrix \(\Sigma\) necessarily have are:
1. Symmetric: The covariance matrix \(\Sigma\) is always symmetric because the covariance between two random variables \(X_i\) and \(X_j\) is the same as the covariance between \(X_j\) and \(X_i\).
2. Positive Semidefinite: The covariance matrix \(\Sigma\) is always positive semidefinite because for any vector \(v\) in \(\mathbb{R}^d\), we have \(v^T\Sigma v \geq 0\). This property ensures that the variances and covariances are non-negative.
3. Orthogonal: The covariance matrix \(\Sigma\) does not necessarily have to be orthogonal. The orthogonality of a matrix refers to its columns or rows being mutually orthogonal unit vectors. Covariance matrices can have non-zero off-diagonal elements, indicating that the variables are correlated.
4. Positive Definite: The covariance matrix \(\Sigma\) may or may not be positive definite. A covariance matrix is positive definite if and only if all its eigenvalues are positive. It is positive semidefinite if and only if all its eigenvalues are non-negative. So, while positive semidefiniteness is guaranteed, positive definiteness depends on the specific correlations and variances of the random variables.
The covariance matrix \Sigma necessarily has the following properties:
1. Symmetric: Yes, \Sigma is symmetric. This means that for any i and j, the (i, j)-th element of \Sigma is equal to the (j, i)-th element.
2. Positive Semidefinite: Yes, \Sigma is positive semidefinite. This means that for any vector \mathrm{{\boldsymbol v}} \in \mathbb{R}^d, \mathrm{{\boldsymbol v}}^T \Sigma \mathrm{{\boldsymbol v}} \geq 0.
3. Positive Definite: Not necessarily. The matrix \Sigma may be positive definite or positive semidefinite, depending on the properties of the random vector \mathrm{{\boldsymbol X}}. However, it is not necessarily positive definite.
4. Orthogonal: No, \Sigma is not necessarily orthogonal. Orthogonal matrices are square matrices with orthogonal columns or rows, but the covariance matrix \Sigma is not necessarily square or orthogonal.
Therefore, the properties that necessarily hold for the covariance matrix \Sigma are Symmetric and Positive Semidefinite.
To determine which properties the covariance matrix \(\Sigma\) necessarily has, let's first define the properties:
1. Symmetric: A matrix is symmetric if it is equal to its own transpose. For every entry \(a_{ij}\) in the matrix, it must be equal to the entry \(a_{ji}\).
2. Positive Semidefinite: A matrix is positive semidefinite if all its eigenvalues are non-negative. In other words, for any non-zero vector \(v\), \(v^T\Sigma v \geq 0\).
3. Positive Definite: A matrix is positive definite if all its eigenvalues are strictly positive. In other words, for any non-zero vector \(v\), \(v^T\Sigma v > 0\).
4. Orthogonal: A matrix is orthogonal if its columns are mutually orthogonal (perpendicular) unit vectors.
Now, let's analyze each property in relation to the covariance matrix \(\Sigma\):
1. Symmetric: The covariance matrix \(\Sigma\) is always symmetric. This is because covariance is a measure of how variables change together, and this relationship is symmetric - the covariance between variables \(X_i\) and \(X_j\) is the same as the covariance between \(X_j\) and \(X_i\).
2. Positive Semidefinite: The covariance matrix \(\Sigma\) is positive semidefinite. This property arises from the definition of covariance, where all variances are non-negative. Since the diagonal entries of \(\Sigma\) represent variances, all eigenvalues of \(\Sigma\) must be non-negative.
3. Positive Definite: The covariance matrix \(\Sigma\) may or may not be positive definite. It depends on the specific random vector \(\mathrm{{\boldsymbol X}}\). If \(\mathrm{{\boldsymbol X}}\) is a degenerate random vector (has a linear dependency), then \(\Sigma\) will have at least one eigenvalue equal to zero, making it not positive definite.
4. Orthogonal: The covariance matrix \(\Sigma\) generally does not have to be orthogonal. Orthogonal matrices represent rotations or reflections, which do not capture the linear relationship between variables measured by covariance.
Therefore, the properties that the covariance matrix \(\Sigma\) necessarily has are:
- Symmetric
- Positive Semidefinite
The property that may or may not apply is:
- Positive Definite
The property that typically does not apply is:
- Orthogonal