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Real symmetric matrices are diagonalizable by orthogonal matrices; i.e., given a real symmetric matrix , is diagonal for some orthogonal matrix . More generally, matrices are diagonalizable by unitary matrices if and only if they are normal .
Every real symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix. If and are real symmetric matrices that commute, then they can be simultaneously diagonalized by an orthogonal matrix: [2] there exists a basis of such that every element of the basis is an eigenvector for both and . Every real symmetric matrix is ...
Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the spectral theorem.
In the case that A is identified with a Hermitian matrix, the matrix of A * is equal to its conjugate transpose. (If A is a real matrix, then this is equivalent to A T = A, that is, A is a symmetric matrix.) This condition implies that all eigenvalues of a Hermitian map are real: To see this, it is enough to apply it to the case when x = y is ...
In linear algebra, an orthogonal diagonalization of a normal matrix (e.g. a symmetric matrix) is a diagonalization by means of an orthogonal change of coordinates. [1]The following is an orthogonal diagonalization algorithm that diagonalizes a quadratic form q(x) on n by means of an orthogonal change of coordinates X = PY.
In case of a symmetric matrix it is the largest absolute value of its eigenvectors and thus equal to its spectral radius. Condition number The condition number of a nonsingular matrix is defined as = ‖ ‖ ‖ ‖. In case of a symmetric matrix it is the absolute value of the quotient of the largest and smallest eigenvalue.
Recall that a Hermitian (or real symmetric) matrix is diagonalizable with only real eigenvalues. It can be shown that, for a given matrix, the Rayleigh quotient reaches its minimum value λ min {\displaystyle \lambda _{\min }} (the smallest eigenvalue of M {\displaystyle M} ) when x {\displaystyle x} is v min {\displaystyle v_{\min }} (the ...
This is because any function of a non-defective matrix acts directly on each of its eigenvalues, and the conjugate transpose of its spectral decomposition is , where is the diagonal matrix of eigenvalues. Likewise, if two normal matrices commute and are therefore simultaneously diagonalizable, any operation between these matrices also acts on ...