symmetric-matrices.wiki (1598B)
1 == Symmetric matrices == 2 symmetric if $A^T = A$ (also has to be square) 3 4 === Diagonalization of symmetric matrices === 5 If A is symmetric, any two eigenvectors from two different eigenspaces are orthogonal. 6 7 An n×n matrix is orthogonally diagonalizable iff A is symmetric. 8 9 An n×n matrix A: 10 * has n real eigenvalues, including multiplicities 11 * is orthogonally diagonalizable 12 * dimension of eigenspace for each eigenvalue λ == multiplicity of λ as root of the characteristic equation ($\det (A-\lambda I) = 0$) 13 * eigenspaces are mutually orthogonal (i.e. eigenvectors corresponding to different eigenvalues are orthogonal) 14 15 === Singular value decomposition === 16 singular values: square roots of eigenvalues of $A^T A$, denoted by $\sigma_1, \dots, \sigma_n$ in ascending order. They are also the lengths of vectors $Av_1, \dots, Av_n$. 17 18 Suppose $\{v_1, \dots, v_n\}$ is an orthonormal basis for $\Re^n$ consisting of eigenvectors of $A^T A$ in ascending order, and suppose A has r nonzero singular values. 19 * Then $\{Av_1, \dots, Av_n\}$ is orthogonal basis for Col A, and rank == r. 20 * Then there exists Σ matrix m×n for which diagonal entries are first r singular values of A, and there exist matrices U (orthogonal, m²) and V (orthogonal, n²) such that $A = U \Sigma V T$. 21 * Col U are "left singular vectors" of A, Col V are "right singular vectors" of A. 22 23 Let A be n², then the fact that "A is invertible" means that: 24 * $(\text{Col} A)^\perp = \{ 0 \}$ 25 * $(\text{Nul} A)^\perp = \Re^n$ 26 * $\text{Row} A = \Re^n$ 27 * A has n nonzero singular values