What is a Hermitian Matrix?
A Hermitian matrix is a special type of square matrix that is equal to its own conjugate transpose:
\[ A = A^H \]
Where \( A^H \) is the conjugate transpose of \( A \) (also known as the adjoint), defined as:
\[ A^H = \overline{A}^T \]
Here:
- \( A^T \) is the transpose of \( A \), where rows become columns.
- \( \overline{A} \) denotes the complex conjugate of the elements of \( A \).
For real-valued matrices, the Hermitian property simplifies to the condition that the matrix is equal to its transpose:
\[ A = A^T \]
Meaning it is a symmetric matrix.
Key Properties of Hermitian Matrices
The properties of Hermitian matrices are particularly useful in the context of Eigenvalue Decomposition:
Real Eigenvalues
The eigenvalues of a Hermitian matrix are always real numbers, which is crucial for numerical stability. This is why when performing Eigenvalue Decomposition on a covariance matrix (which is Hermitian), the eigenvalues will be real and can be interpreted in terms of the variance captured by each principal component.
Orthogonal Eigenvectors
The eigenvectors corresponding to distinct eigenvalues of a Hermitian matrix are orthogonal to each other. This property ensures that, when we perform eigenvalue decomposition on the covariance matrix, the resulting eigenvectors will be orthogonal. This is especially important in applications like Principal Component Analysis (PCA), where the orthogonality of the eigenvectors corresponds to the fact that the principal components (the directions of maximum variance) are independent of each other.
Diagonalizable
Hermitian matrices are diagonalizable, meaning they can be written as:
where \( V \) is a matrix of eigenvectors, and \( \Lambda \) is a diagonal matrix of eigenvalues. This property ensures that we can represent the covariance matrix in terms of its eigenvectors and eigenvalues, making it possible to reduce dimensions by discarding small eigenvalues (which correspond to noise).
The Role of Hermitian Matrices in Noise Reduction
In the context of noise reduction via Eigenvalue Decomposition (EVD), the covariance matrix's Hermitian nature is key for several reasons:
Real Eigenvalues
The covariance matrix being Hermitian guarantees that the eigenvalues will be real and non-negative (since they correspond to variance). This allows us to interpret these eigenvalues as quantities representing the amount of signal (variance) along each principal component direction.
Diagonalization
Since the covariance matrix is Hermitian, we can diagonalize it using its orthonormal eigenvectors. This means we can decompose the data into components that are independent of each other (principal components), which is the basis of PCA and other dimensionality reduction techniques.
Principal Component Analysis (PCA)
PCA is a method that relies on the eigenvalue decomposition of the covariance matrix. Because the covariance matrix is Hermitian, the eigenvectors form an orthogonal basis, and the corresponding eigenvalues tell us how much of the variance (signal) is captured along each eigenvector direction.
By selecting the eigenvectors with the largest eigenvalues, we can effectively reduce noise by ignoring the components associated with smaller eigenvalues (which often correspond to noise).
Noise Reduction
When reducing noise, we discard the directions with small eigenvalues (corresponding to small variance and likely noise), and keep the directions with large eigenvalues (the principal components, which correspond to the significant signal in the data).
The Hermitian property ensures that these discarded components are orthogonal to the retained components, preserving the structure of the data while eliminating noise.