notation are equivalent to numerator layout with the denominators written without the transpose. Keep in mind that various authors use different combinations of numerator and denominator layouts for different types of derivatives, and there is no guarantee that an author will consistently use either numerator or denominator layout for all types. Match up the formulas below with those quoted in the source to determine the layout used for that particular type of derivative, but be careful not to assume that derivatives of other types necessarily follow the same kind of layout. When taking derivatives with an aggregate (vector or matrix) denominator in order to find a maximum or minimum of the aggregate, it should be kept in mind that using numerator layout will produce results that are transposed with respect to the aggregate. For example, in attempting to find the maximum likelihood estimate of a multivariate normal distribution using matrix calculus, if the domain is a k×1 column vector, then the result using the numerator layout will be in the form of a 1×k row vector. Thus, either the results should be transposed at the end or the denominator layout (or mixed layout) should be used. {|class="wikitable" |+ Result of differentiating various kinds of aggregates with other kinds of aggregates ! colspan=2 rowspan=2 | ! colspan=2 | Scalar y ! colspan=2 | Column vector y (size m×1) ! colspan=2 | Matrix Y (size m×n) |- ! Notation !! Type ! Notation !! Type ! Notation !! Type |- ! rowspan=2 | Scalar x ! Numerator | rowspan=2 style="text-align:center;" | | rowspan=2 | Scalar | rowspan=2 style="text-align:center;" | | Size-m column vector | rowspan=2 style="text-align:center;" | | m×n matrix |- ! Denominator | Size-m row vector | |- ! rowspan=2 | Column vector x(size n×1) ! Numerator | rowspan=2 style="text-align:center;" | | Size-n row vector | rowspan=2 style="text-align:center;" | | m×n matrix | rowspan=2 style="text-align:center;" | | rowspan=2 | |- ! Denominator | Size-n column vector | n×m matrix |- ! rowspan=2 | Matrix X(size p×q) ! Numerator | rowspan=2 style="text-align:center;" | | q×p matrix | rowspan=2 style="text-align:center;" | | rowspan=2 | | rowspan=2 style="text-align:center;" | | rowspan=2 | |- ! Denominator | p×q matrix |} The results of operations will be transposed when switching between numerator-layout and denominator-layout notation. Numerator-layout notation Using numerator-layout notation, we have: The following definitions are only provided in numerator-layout notation: Denominator-layout notation Using denominator-layout notation,