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The th column of an identity matrix is the unit vector, a vector whose th entry is 1 and 0 elsewhere. The determinant of the identity matrix is 1, and its trace is . The identity matrix is the only idempotent matrix with non-zero determinant. That is, it is the only matrix such that:
Let A be an m × n matrix. Let the column rank of A be r, and let c 1, ..., c r be any basis for the column space of A. Place these as the columns of an m × r matrix C. Every column of A can be expressed as a linear combination of the r columns in C. This means that there is an r × n matrix R such that A = CR.
Every finite-dimensional matrix has a rank decomposition: Let be an matrix whose column rank is . Therefore, there are r {\textstyle r} linearly independent columns in A {\textstyle A} ; equivalently, the dimension of the column space of A {\textstyle A} is r {\textstyle r} .
The map , representing scalar multiplication as a sum of outer products. The generalized Kronecker delta or multi-index Kronecker delta of order 2 p {\displaystyle 2p} is a type ( p , p ) {\displaystyle (p,p)} tensor that is completely antisymmetric in its p {\displaystyle p} upper indices, and also in its p {\displaystyle p} lower indices.
Composable differentiable functions f : R n → R m and g : R m → R k satisfy the chain rule, namely () = (()) for x in R n. The Jacobian of the gradient of a scalar function of several variables has a special name: the Hessian matrix , which in a sense is the " second derivative " of the function in question.
A common case is finding the inverse of a low-rank update A + UCV of A (where U only has a few columns and V only a few rows), or finding an approximation of the inverse of the matrix A + B where the matrix B can be approximated by a low-rank matrix UCV, for example using the singular value decomposition.
For an orthogonal matrix R, note that det R T = det R implies (det R) 2 = 1, so that det R = ±1. The subgroup of orthogonal matrices with determinant +1 is called the special orthogonal group, denoted SO(3). Thus every rotation can be represented uniquely by an orthogonal matrix with unit determinant.
In Python NumPy arrays implement the flatten method, [note 1] while in R the desired effect can be achieved via the c() or as.vector() functions or, more efficiently, by removing the dimensions attribute of a matrix A with dim(A) <- NULL. In R, function vec() of package 'ks' allows vectorization and function vech() implemented in both packages ...