Question
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
Which of the following properties does \Sigma necessarily have? (Choose all that apply.)
Symmetric
Positive Semidefinite
Positive Definite
Orthogonal
Answers
GPT 3.5
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.
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.
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