A pencil (ˈpɛnsəl) is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage, and keeps it from marking the user's hand. Pencils create marks by physical abrasion, leaving a trail of solid core material that adheres to a sheet of paper or other surface. They are distinct from pens, which dispense liquid or gel ink onto the marked surface. Most pencil cores are made of graphite powder mixed with a clay binder.
In linear algebra, an eigenvector (ˈaɪgənˌvɛktər) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a constant factor when that linear transformation is applied to it. The corresponding eigenvalue, often represented by , is the multiplying factor. Geometrically, a transformation matrix rotates, stretches, or shears the vectors it acts upon. The eigenvectors for a linear transformation matrix are the set of vectors that are only stretched, with no rotation or shear.
A mechanical pencil, also clutch pencil, is a pencil with a replaceable and mechanically extendable solid pigment core called a "lead" ˈlɛd. The lead, often made of graphite, is not bonded to the outer casing, and can be mechanically extended as its point is worn away as it is being used. The vast majority of mechanical pencils have erasers. Other names include: microtip pencil, automatic pencil, drafting pencil, technical pencil, draughting pencil, click pencil (generally refers to a specific brand), pump pencil, leadholder, Pacer (Australian English, ca.
In linear algebra, a generalized eigenvector of an matrix is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector. Let be an -dimensional vector space and let be the matrix representation of a linear map from to with respect to some ordered basis. There may not always exist a full set of linearly independent eigenvectors of that form a complete basis for . That is, the matrix may not be diagonalizable.
In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. 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. Eigenvalue, eigenvector and eigenspace A (nonzero) vector v of dimension N is an eigenvector of a square N × N matrix A if it satisfies a linear equation of the form for some scalar λ.
