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The inverse of an upper triangular matrix, if it exists, is upper triangular. The product of an upper triangular matrix and a scalar is upper triangular. Together these facts mean that the upper triangular matrices form a subalgebra of the associative algebra of square matrices for a given size.
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.
It is called an identity matrix because multiplication with it leaves a matrix unchanged: = = for any m-by-n matrix A. A nonzero scalar multiple of an identity matrix is called a scalar matrix. If the matrix entries come from a field, the scalar matrices form a group, under matrix multiplication, that is isomorphic to the multiplicative group ...
If is a triangular matrix, i.e. =, whenever > or, alternatively, whenever <, then its determinant equals the product of the diagonal entries: () = = =. Indeed, such a matrix can be reduced, by appropriately adding multiples of the columns with fewer nonzero entries to those with more entries, to a diagonal matrix (without changing the determinant).
A special case is a diagonal matrix, with arbitrary numbers ,, … along the diagonal: the axes of scaling are then the coordinate axes, and the transformation scales along each axis by the factor . In uniform scaling with a non-zero scale factor, all non-zero vectors retain their direction (as seen from the origin), or all have the direction ...
Related: the LDU decomposition is =, where L is lower triangular with ones on the diagonal, U is upper triangular with ones on the diagonal, and D is a diagonal matrix. Related: the LUP decomposition is =, where L is lower triangular, U is upper triangular, and P is a permutation matrix.
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
The identity matrix commutes with all matrices. Jordan blocks commute with upper triangular matrices that have the same value along bands. If the product of two symmetric matrices is symmetric, then they must commute. That also means that every diagonal matrix commutes with all other diagonal matrices. [9] [10] Circulant matrices commute.