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An identity matrix of any size, or any multiple of it is a diagonal matrix called a scalar matrix, for example, []. In geometry , a diagonal matrix may be used as a scaling matrix , since matrix multiplication with it results in changing scale (size) and possibly also shape ; only a scalar matrix results in uniform change in scale.
The 4-wavevector is the 4-gradient of the negative phase (or the negative 4-gradient of the phase) of a wave in Minkowski Space: [6]: 387 = = (,) = [] = [] This is mathematically equivalent to the definition of the phase of a wave (or more specifically a plane wave ): K ⋅ X = ω t − k → ⋅ x → = − Φ {\displaystyle \mathbf {K} \cdot ...
The binary matrix with ones on the anti-diagonal, and zeroes everywhere else. a ij = δ n+1−i,j: A permutation matrix. Hilbert matrix: a ij = (i + j − 1) −1. A Hankel matrix. Identity matrix: A square diagonal matrix, with all entries on the main diagonal equal to 1, and the rest 0. a ij = δ ij: Lehmer matrix: a ij = min(i, j) ÷ max(i, j).
Thus, for example, the product of a 1 × n matrix and an n × 1 matrix, which is formally a 1 × 1 matrix, is often said to be a scalar. The real component of a quaternion is also called its scalar part. The term scalar matrix is used to denote a matrix of the form kI where k is a scalar and I is the identity matrix.
The next type of row operation on a matrix A multiplies all elements on row i by m where m is a non-zero scalar (usually a real number). The corresponding elementary matrix is a diagonal matrix, with diagonal entries 1 everywhere except in the i th position, where it is m.
A scalar function that depends entirely on the principal invariants of a tensor is objective, i.e., independent of rotations of the coordinate system. This property is commonly used in formulating closed-form expressions for the strain energy density , or Helmholtz free energy , of a nonlinear material possessing isotropic symmetry.
An example of a matrix in Jordan normal form. The grey blocks are called Jordan blocks. The eigendecomposition or diagonalization expresses A as a product VDV −1, where D is a diagonal matrix and V is a suitable invertible matrix. [52] If A can be written in this form, it is called diagonalizable.
In geometric algebra, a multivector is defined to be the sum of different-grade k-blades, such as the summation of a scalar, a vector, and a 2-vector. [17] A sum of only k-grade components is called a k-vector, [18] or a homogeneous multivector. [19] The highest grade element in a space is called a pseudoscalar.