Vector space Definition (over reals) A set X is called a vector space
... orthonormal A set of vectors {x1 , . . . , xn } is orthonormal if hxi , xj i = δij where δij = 1 if i = j, 0 otherwise. Eigenvalues and eigenvectors Definition Given a matrix M , the real value λ and (non-zero) vector x are an eigenvalue and corresponding eigenvector of M if M x = λx Spectral decomp ...
... orthonormal A set of vectors {x1 , . . . , xn } is orthonormal if hxi , xj i = δij where δij = 1 if i = j, 0 otherwise. Eigenvalues and eigenvectors Definition Given a matrix M , the real value λ and (non-zero) vector x are an eigenvalue and corresponding eigenvector of M if M x = λx Spectral decomp ...
1 The Covariance Matrix
... subspace and we can select orthogonal vectors spanning this subspace. So there always exists an orhtonormal set of eigenvectors of Σ. It is often convenient to work in an orthonormal coordinate system where the coordinate axes are eigenvectors of Σ. In this coordinte system we have that Σ is a diag ...
... subspace and we can select orthogonal vectors spanning this subspace. So there always exists an orhtonormal set of eigenvectors of Σ. It is often convenient to work in an orthonormal coordinate system where the coordinate axes are eigenvectors of Σ. In this coordinte system we have that Σ is a diag ...
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eigenvalue problem
... equation in l, known as the characteristic equation corresponding to the matrix A, and its left-hand side is an nth degree polynomial known as the characteristic polynomial. Let us consider Eq.(2) to have been solved for the eigenvalues l1,l2,…,lk (1 ≦ k ≦ n). Next, set l = l1 in Eq. (1). Since det( ...
... equation in l, known as the characteristic equation corresponding to the matrix A, and its left-hand side is an nth degree polynomial known as the characteristic polynomial. Let us consider Eq.(2) to have been solved for the eigenvalues l1,l2,…,lk (1 ≦ k ≦ n). Next, set l = l1 in Eq. (1). Since det( ...
