of X || ||| |- style="border-top: 3px solid;" | || || colspan=2| |- | U = U(X), V = V(X) || || colspan=2| |- | a is not a function of X, U = U(X) || || colspan=2| |- | g(X) is any polynomial with scalar coefficients, or any matrix function defined by an infinite polynomial series (e.g. eX, sin(X), cos(X), ln(X), etc. using a Taylor series); g(x) is the equivalent scalar function, g′(x) is its derivative, and g′(X) is the corresponding matrix function || || || |- | A is not a function of X || || || |- | A is not a function of X || || || |- | A is not a function of X || || || |- | A is not a function of X || || || |- | A, B are not functions of X || || || |- | A, B, C are not functions of X || || || |- | n is a positive integer || || || |- | A is not a function of X, n is a positive integer || || || |- | || || || |- | || || || |- style="border-top: 3px solid;" | || || || |- | a is not a function of X || || || |- | A, B are not functions of X || || || |- | n is a positive integer || || || |- | (see pseudo-inverse) || || || |- | (see pseudo-inverse) || || || |- | A is not a function of X,X is square and invertible || || || |- | A is not a function of X,X is non-square,A is symmetric || || || |- | A is not a function of X,X is non-square,A is non-symmetric || | | |} Matrix-by-scalar identities {|class="wikitable" style="text-align: center;" |+ Identities: matrix-by-scalar ! scope="col" width="175" | Condition ! scope="col" width="100" | Expression ! scope="col" width="100" | Numerator layout, i.e. by Y |- | U = U(x) || || |- | A, B are not functions of x, U = U(x) || || |- | U = U(x), V = V(x) || || |- | U = U(x), V = V(x) || || |- | U = U(x), V = V(x) || || |- | U = U(x), V = V(x) || || |- | U = U(x) || || |- | U