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The Spectral Theorem states that any real symmetric matrix can be diagonalized by an orthogonal matrix, meaning it can be expressed in the form $QDQ^T$, where $Q$ is an orthogonal matrix and $D$ is a diagonal matrix.

In quantum mechanics, the Spectral Theorem assists in defining the continuous spectrum of an observable, which corresponds to the set of non-discrete eigenvalues of the associated Hermitian operator.

The Spectral Theorem for normal matrices, which includes Hermitian, unitary, and skew-Hermitian matrices, asserts that they can be diagonalized by a unitary matrix.

The Spectral Radius of a matrix, which is the maximum absolute value of its eigenvalues, can be determined using the Spectral Theorem.

The Spectral Theorem for self-adjoint operators in Hilbert spaces implies that such operators can be expressed as an integral over their spectrum, with a projection-valued measure.

The Spectral Mapping Theorem, a corollary to the Spectral Theorem, states that for any polynomial $p$, the eigenvalues of $p(A)$ are given by $p(\lambda)$, where $\lambda$ are the eigenvalues of $A$.

While not a direct implication of the Spectral Theorem, the Perron-Frobenius theorem for positive matrices complements the Spectral Theorem by addressing the existence and uniqueness of a largest positive eigenvalue.

A conclusion of the Spectral Theorem is that all the eigenvalues of a real symmetric matrix are real numbers.

For compact self-adjoint operators on a Hilbert space, the Spectral Theorem shows that the space can be decomposed into a direct sum of eigenspaces corresponding to the non-zero eigenvalues.

The Spectral Theorem implies that the eigenvectors corresponding to distinct eigenvalues of a real symmetric matrix are orthogonal to each other.

In the context of bounded operators on a Hilbert space, the Spectral Theorem states that any normal operator can be decomposed into a set of eigenvectors that form an orthonormal basis, with corresponding eigenvalues that can be complex numbers.

The Rayleigh Quotient, defined as $R(x) = \frac{x^TAx}{x^Tx}$ for a non-zero vector $x$, becomes the eigenvalue when $x$ is an eigenvector of a real symmetric matrix, as per the Spectral Theorem.

The Spectral Decomposition is a result derived from the Spectral Theorem, which expresses a matrix as a sum of rank-one matrices multiplied by their corresponding eigenvalues: $A = \lambda_1v_1v_1^T + \lambda_2v_2v_2^T + \cdots + \lambda_nv_nv_n^T$.

The Spectral Theorem is foundational in PCA, which uses eigendecomposition of the covariance matrix to identify the principal components, revealing the underlying structure of the data.