Search results
Results From The WOW.Com Content Network
In linear algebra, the identity matrix of size is the square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation , the object remains unchanged by the transformation.
A square matrix of order 4. The entries form the main diagonal of a square matrix. For instance, the main diagonal of the 4×4 matrix above contains the elements a 11 = 9, a 22 = 11, a 33 = 4, a 44 = 10. In mathematics, a square matrix is a matrix with the same number of rows and columns.
The identity matrix I n of size n is the n-by-n matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, for example, = [], = [], = [] It is a square matrix of order n, and also a special kind of diagonal matrix.
A square matrix derived by applying an elementary row operation to the identity matrix. Equivalent matrix: A matrix that can be derived from another matrix through a sequence of elementary row or column operations. Frobenius matrix: A square matrix in the form of an identity matrix but with arbitrary entries in one column below the main diagonal.
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q T Q = Q Q T = I , {\displaystyle Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I,} where Q T is the transpose of Q and I is the identity matrix .
That is, multiplication by the matrix is an involution if and only if =, where is the identity matrix. Involutory matrices are all square roots of the identity matrix. This is a consequence of the fact that any invertible matrix multiplied by its inverse is the identity.
Another way to find the square root of an n × n matrix A is the Denman–Beavers square root iteration. [8] Let Y 0 = A and Z 0 = I, where I is the n × n identity matrix. The iteration is defined by
In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself. [1] [2] That is, the matrix is idempotent if and only if =. For this product to be defined, must necessarily be a square matrix.