symmetric-matrices.md (1605B)
1 +++ 2 title = 'Symmetric matrices' 3 template = 'page-math.html' 4 +++ 5 6 # Symmetric matrices 7 symmetric if $A^T = A$ (also has to be square) 8 9 ## Diagonalization of symmetric matrices 10 If A is symmetric, any two eigenvectors from two different eigenspaces are orthogonal. 11 12 An n×n matrix is orthogonally diagonalizable iff A is symmetric. 13 14 An n×n matrix A: 15 * has n real eigenvalues, including multiplicities 16 * is orthogonally diagonalizable 17 * dimension of eigenspace for each eigenvalue λ == multiplicity of λ as root of the characteristic equation ($\det (A-\lambda I) = 0$) 18 * eigenspaces are mutually orthogonal (i.e. eigenvectors corresponding to different eigenvalues are orthogonal) 19 20 ## Singular value decomposition 21 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 $Av_1, \dots, Av_n$. 22 23 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. 24 * Then $\{Av_1, \dots, Av_n\}$ is orthogonal basis for Col A, and rank == r. 25 * 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 $A = U \Sigma V T$. 26 * Col U are "left singular vectors" of A, Col V are "right singular vectors" of A. 27 28 Let A be n², then the fact that "A is invertible" means that: 29 * $(\text{Col} A)^\perp = \{ 0 \}$ 30 * $(\text{Nul} A)^\perp = \Re^n$ 31 * $\text{Row} A = \Re^n$ 32 * A has n nonzero singular values