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The Matrix Form of the Chain Rule for Compositions of Differentiable Functions from Rn to Rm. Recall from The Chain Rule for Compositions of Differentiable Functions from Rn to Rm page that if is open, , , and if is another function such that the composition is well defined then if is differentiable at with total derivative and is differentiable at with total derivative then is differentiable at and: vec(A) The vector-version of the matrix A (see Sec. 10.2.2) sup Supremum of a set jjAjj Matrix norm (subscript if any denotes what norm) AT Transposed matrix A TThe inverse of the transposed and vice versa, A T = (A 1)T = (A ) . A Complex conjugated matrix AH Transposed and complex conjugated matrix (Hermitian) A B Hadamard (elementwise) product A The Chain Rule As a motivation for the chain rule, consider the function f (x) = (1+ x2) 10. Since f (x) is a polynomial function, we know from previous pages that f ' (x) exists. derivative matrix in terms of the new, primed variables û2J ûx'2 = (û ûx') T (û ûx') J, (3.3) AT (û ûx) T (û ûx) AJ, (3.4) = AT û2J ûx2 A. (3.5) 4. Chain Rule for Vector Functions (First Derivative) If the function itself is a vector, f (x), then the derivative is a matrix ûf ûx = ( ), (4.1) ûf1 /ûx1 ûf1 /ûx2 ûf1 /ûxn ûf2 ...

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The Matrix Form of the Chain Rule for Compositions of Differentiable Functions from Rn to Rm. Recall from The Chain Rule for Compositions of Differentiable Functions from Rn to Rm page that if is open, , , and if is another function such that the composition is well defined then if is differentiable at with total derivative and is differentiable at with total derivative then is differentiable at and: Rule Comments (AB)T= BTATorder is reversed, everything is transposed (aTBc)T= c B a as above aTb = b a (the result is a scalar, and the transpose of a scalar is itself) (A+ B)C = AC+ BC multiplication is distributive (a+ b)TC = aTC+ bTC as above, with vectors AB 6= BA multiplication is not commutative 2 Common vector derivatives

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Then, DF= 2xy3z43x2y2z44x2y3z3. exsin(yz) zexcos(yz) yexcos(yz) : 3. The Chain Rule Stating the Chain Rule in terms of the derivative matrices is strikingly similar to the well-known (f g)0(x) = f0(g(x)) g0(x). The main di erence is that we use matrix multiplication! The chain rule tells us how to find the derivative of a composite function. Brush up on your knowledge of composite functions, and learn how to apply the chain rule correctly. If you're seeing this message, it means we're having trouble loading external resources on our website. Week 2 of the Course is devoted to the main concepts of differentiation, gradient and Hessian. Of special attention is the chain rule. Also students will understand economic applications of the gradient. The multivariable chain rule is more often expressed in terms of the gradient and a vector-valued derivative. This makes it look very analogous to the single... The chain rule is a rule for differentiating compositions of functions. In the following discussion and solutions the derivative of a function h(x) will be denoted by or h'(x) . Most problems are average. A few are somewhat challenging. The chain rule states formally that . However, we rarely use this formal approach when applying the chain ...

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vec(A) The vector-version of the matrix A (see Sec. 10.2.2) sup Supremum of a set jjAjj Matrix norm (subscript if any denotes what norm) AT Transposed matrix A TThe inverse of the transposed and vice versa, A T = (A 1)T = (A ) . A Complex conjugated matrix AH Transposed and complex conjugated matrix (Hermitian) A B Hadamard (elementwise) product A The chain rule is a rule for differentiating compositions of functions. In the following discussion and solutions the derivative of a function h(x) will be denoted by or h'(x) . Most problems are average. A few are somewhat challenging. The chain rule states formally that . However, we rarely use this formal approach when applying the chain ... chain rule. By doing all of these things at the same time, we are more likely to make errors, at least until we have a lot of experience. 1.1 Expanding notation into explicit sums and equations for each component In order to simplify a given calculation, it is often useful to write out the explicit formula for A superscript T denotes the matrix transpose operation; for example, AT denotes the transpose of A. Similarly, if A has an inverse it will be denoted by A-1. The determinant of A will be denoted by either jAj or det(A). Similarly, the rank of a matrix A is denoted by rank(A). An identity matrix will be denoted by I, and 0 will denote a null matrix.

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Rather than the chain rule, let's tackle the problem using differentials. Let's use the convention that an upppercase letter is a matrix, lowercase is a column vector, and a greek letter is a scalar. Now let's define some variables and their differentials z = W 1 x + b 1 ⟹ d z = d W 1 x h = r e l u (z) ⟹ d h = s t e p (z) ⊙ d z = s ⊙ d z Aug 12, 2020 · Chain Rules for One or Two Independent Variables Recall that the chain rule for the derivative of a composite of two functions can be written in the form d dx(f(g(x))) = f′ (g(x))g′ (x). In this equation, both f(x) and g(x) are functions of one variable. Aug 25, 2020 · Given an array p [] which represents the chain of matrices such that the ith matrix Ai is of dimension p [i-1] x p [i]. We need to write a function MatrixChainOrder () that should return the minimum number of multiplications needed to multiply the chain. As you can probably imagine, the multivariable chain rule generalizes the chain rule from single variable calculus. The single variable chain rule tells you how to take the derivative of the composition of two functions: \dfrac {d} {dt}f (g (t)) = \dfrac {df} {dg} \dfrac {dg} {dt} = f' (g (t))g' (t) dtd f (g(t)) = dgdf

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The Matrix Form of the Chain Rule for Compositions of Differentiable Functions from Rn to Rm. Recall from The Chain Rule for Compositions of Differentiable Functions from Rn to Rm page that if is open, , , and if is another function such that the composition is well defined then if is differentiable at with total derivative and is differentiable at with total derivative then is differentiable at and: Rule f(x) Scalar derivative notation with respect to x Example Constant c 0 d dx 99 = 0 Multiplication by constant cf cdf dx d dx 3x= 3 Power Rule xn nxn 1 d dx x 3 = 3x2 Sum Rule f+ g df dx + dg dx d dx (x 2 + 3x) = 2x+ 3 Di erence Rule f g df dx dg dx d dx (x 2 3x) = 2x 3 Product Rule fg fdg dx + df dx g d dx x 2x= x2 + x2x= 3x Chain Rule f(g ... Then, DF= 2xy3z43x2y2z44x2y3z3. exsin(yz) zexcos(yz) yexcos(yz) : 3. The Chain Rule Stating the Chain Rule in terms of the derivative matrices is strikingly similar to the well-known (f g)0(x) = f0(g(x)) g0(x). The main di erence is that we use matrix multiplication! The chain rule tells us how to find the derivative of a composite function. Brush up on your knowledge of composite functions, and learn how to apply the chain rule correctly. If you're seeing this message, it means we're having trouble loading external resources on our website.

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Feb 27, 2018 · Vector chain rule :-Vector chain rule for vectors of functions and a single parameter mirrors the single-variable chain rule. If y = f ( g ( x )) and x is a vector . That last equation is the chain rule in this gen-eralization. The formal proof depends on the ordi-nary de nition of derivative and the usual proper-ties of limits, but as this is a form of the chain rule, the proof has a lot of details. The rst step for general m. Of course, this generalizes to df dt = @f @x 1 dx 1 dt + @f @x 2 dx 2 dt + + @f ...

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The multivariable chain rule is more often expressed in terms of the gradient and a vector-valued derivative. This makes it look very analogous to the single... The Chain Rule As a motivation for the chain rule, consider the function f (x) = (1+ x2) 10. Since f (x) is a polynomial function, we know from previous pages that f ' (x) exists. A superscript T denotes the matrix transpose operation; for example, AT denotes the transpose of A. Similarly, if A has an inverse it will be denoted by A-1. The determinant of A will be denoted by either jAj or det(A). Similarly, the rank of a matrix A is denoted by rank(A). An identity matrix will be denoted by I, and 0 will denote a null matrix.

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This rule, which became final in September 2015, requires food facilities to have a food safety plan in place that includes an analysis of hazards and risk-based preventive controls to minimize or ... use the chain rule. The reason is most interesting problems in physics and engineering are equations involving partial derivatives, that is partial di erential equations. These equations normally have physical interpretations and are derived from observations and experimenta-tion. With matrix notation J ah = J f( )g · J af. Example 17. If f : R → R, g : R → R and h = g f then (h0(x)) = J xh by Example 14 = J f(x)g · J xf by the Chain Rule = (D 1g(f(x))) · (D 1f(x)) = (g0(f(x))) · (f0(x)) = (g0(f(x))f0(x)) This is the well-known 1-dimensional version of the chain rule. The chain rule from single variable calculus has a direct analogue in multivariable calculus, where the derivative of each function is replaced by its Jacobian matrix, and multiplication is replaced with matrix multiplication.

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a matrix, and the nature of T determine the structure of the result U. For example, ... (the chain rule). We can use expression (4.2) to work these out. For example ... Rule Comments (AB)T= BTATorder is reversed, everything is transposed (aTBc)T= c B a as above aTb = b a (the result is a scalar, and the transpose of a scalar is itself) (A+ B)C = AC+ BC multiplication is distributive (a+ b)TC = aTC+ bTC as above, with vectors AB 6= BA multiplication is not commutative 2 Common vector derivatives

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The chain rule applies in some of the cases, but unfortunately does not apply in matrix-by-scalar derivatives or scalar-by-matrix derivatives (in the latter case, mostly involving the trace operator applied to matrices). In the latter case, the product rule can't quite be applied directly, either, but the equivalent can be done with a bit more work using the differential identities. Chain rule examples: Exponential Functions. Differentiating using the chain rule usually involves a little intuition. The more times you apply the chain rule to different problems, the easier it becomes to recognize how to apply the rule. This section shows how to differentiate the function y = 3x + 1 2 using the chain rule. It again turns out that one does not need matrices to talk about the chain rule but that if one had matrix notation, the matrix notation is particularly convenient for summarizing the chain rule. I have elected to hold off on matrix algebra till the near future because it comes up in a much better motivated way, I think, in terms of these ... Nov 20, 2019 · Now contrast this with the previous problem. In the previous problem we had a product that required us to use the chain rule in applying the product rule. In this problem we will first need to apply the chain rule and when we go to integrate the inside function we’ll need to use the product rule. Here is the chain rule portion of the problem.

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As you can probably imagine, the multivariable chain rule generalizes the chain rule from single variable calculus. The single variable chain rule tells you how to take the derivative of the composition of two functions: \dfrac {d} {dt}f (g (t)) = \dfrac {df} {dg} \dfrac {dg} {dt} = f' (g (t))g' (t) dtd f (g(t)) = dgdf Aug 25, 2020 · Given an array p [] which represents the chain of matrices such that the ith matrix Ai is of dimension p [i-1] x p [i]. We need to write a function MatrixChainOrder () that should return the minimum number of multiplications needed to multiply the chain.

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Rule Comments (AB)T= BTATorder is reversed, everything is transposed (aTBc)T= c B a as above aTb = b a (the result is a scalar, and the transpose of a scalar is itself) (A+ B)C = AC+ BC multiplication is distributive (a+ b)TC = aTC+ bTC as above, with vectors AB 6= BA multiplication is not commutative 2 Common vector derivatives As you can probably imagine, the multivariable chain rule generalizes the chain rule from single variable calculus. The single variable chain rule tells you how to take the derivative of the composition of two functions: \dfrac {d} {dt}f (g (t)) = \dfrac {df} {dg} \dfrac {dg} {dt} = f' (g (t))g' (t) dtd f (g(t)) = dgdf

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The multivariable chain rule is more often expressed in terms of the gradient and a vector-valued derivative. This makes it look very analogous to the single... The multivariable chain rule is more often expressed in terms of the gradient and a vector-valued derivative. This makes it look very analogous to the single...

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derivative matrix in terms of the new, primed variables û2J ûx'2 = (û ûx') T (û ûx') J, (3.3) AT (û ûx) T (û ûx) AJ, (3.4) = AT û2J ûx2 A. (3.5) 4. Chain Rule for Vector Functions (First Derivative) If the function itself is a vector, f (x), then the derivative is a matrix ûf ûx = ( ), (4.1) ûf1 /ûx1 ûf1 /ûx2 ûf1 /ûxn ûf2 ... Formulating the chain rule using the generalized Jacobian yields the same equation as before: for z = f (y) and y = g (x), ∂ z ∂ x = ∂ z ∂ y ∂ y ∂ x. The only difference this time is that ∂ z ∂ x has the shape ( K 1 × . . . × K D z ) × ( M 1 × . . . × M D x ) which is itself formed by the result of a generalized matrix multiplication between the two generalized matrices, ∂ z ∂ y and ∂ y ∂ x . The chain rule is a formula for finding the derivative of a composite function. It uses a variable$ y$ depending on a second variable,$ u$, which in turn depend on a third variable,$ x$. The derivative of any function is the derivative of the function itself, as per the power rule, then the derivative of the inside of the function. May 31, 2018 · With the first chain rule written in this way we can think of \(\eqref{eq:eq1}\) as a formula for differentiating any function of \(x\) and \(y\) with respect to \(\theta \) provided we have \(x = r\cos \theta \) and \(y = r\sin \theta \).

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Apr 04, 2018 · The chain rule thus provides a convenient tool which allows us to correctly find the influence of early layer parameters on the final loss. Local gradients .. Suppose we define a toy computational graph with the following It again turns out that one does not need matrices to talk about the chain rule but that if one had matrix notation, the matrix notation is particularly convenient for summarizing the chain rule. I have elected to hold off on matrix algebra till the near future because it comes up in a much better motivated way, I think, in terms of these ... chain rule. By doing all of these things at the same time, we are more likely to make errors, at least until we have a lot of experience. 1.1 Expanding notation into explicit sums and equations for each component In order to simplify a given calculation, it is often useful to write out the explicit formula for

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That last equation is the chain rule in this gen-eralization. The formal proof depends on the ordi-nary de nition of derivative and the usual proper-ties of limits, but as this is a form of the chain rule, the proof has a lot of details. The rst step for general m. Of course, this generalizes to df dt = @f @x 1 dx 1 dt + @f @x 2 dx 2 dt + + @f ... 4. Chain Rule for Vector Functions (First Derivative) If the function itself is a vector, , then the derivative is a matrix, where the number of components of () is not necessarily the same as the number of components of ().

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Using the notation of matrices of partial derivatives, we can rewrite the one-variable chain rule of equation (1) as (2) D h (t) = D f (g (t)) D g (t). Since matrix multiplication of 1 × 1 matrices is the same as scalar multiplication, this new equation is just equation (1) in disguised form. In calculus, the chain rule is a formula to compute the derivative of a composite function. That is, if f and g are differentiable functions, then the chain rule expresses the derivative of their composite f ∘ g — the function which maps x to

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The Matrix Form of the Chain Rule for Compositions of Differentiable Functions from Rn to Rm. Recall from The Chain Rule for Compositions of Differentiable Functions from Rn to Rm page that if is open, , , and if is another function such that the composition is well defined then if is differentiable at with total derivative and is differentiable at with total derivative then is differentiable at and: a matrix, and the nature of T determine the structure of the result U. For example, ... (the chain rule). We can use expression (4.2) to work these out. For example ... 0.3 \cdot 0.3 + 0.7 \cdot 0.8 = 0.65 0.3⋅0.3+0.7⋅0.8 = 0.65. Since there are only two states in the chain, the process must be on B if it is not on A, and therefore the probability that the process will be on B after 2 moves is. 1 − 0.65 = 0.35. 1 - 0.65 = \boxed {0.35}. 1 −0.65 = 0.35. . Matrix-Chain Multiplication • Let A be an n by m matrix, let B be an m by p matrix, then C = AB is an n by p matrix. • C = AB can be computed in O(nmp) time, using traditional matrix multiplication. • Suppose I want to compute A 1A 2A 3A 4. • Matrix Multiplication is associative, so I can do the multiplication in several diﬀerent ... Rule Comments (AB)T= BTATorder is reversed, everything is transposed (aTBc)T= c B a as above aTb = b a (the result is a scalar, and the transpose of a scalar is itself) (A+ B)C = AC+ BC multiplication is distributive (a+ b)TC = aTC+ bTC as above, with vectors AB 6= BA multiplication is not commutative 2 Common vector derivatives

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Rule f(x) Scalar derivative notation with respect to x Example Constant c 0 d dx 99 = 0 Multiplication by constant cf cdf dx d dx 3x= 3 Power Rule xn nxn 1 d dx x 3 = 3x2 Sum Rule f+ g df dx + dg dx d dx (x 2 + 3x) = 2x+ 3 Di erence Rule f g df dx dg dx d dx (x 2 3x) = 2x 3 Product Rule fg fdg dx + df dx g d dx x 2x= x2 + x2x= 3x Chain Rule f(g ... With matrix notation J ah = J f( )g · J af. Example 17. If f : R → R, g : R → R and h = g f then (h0(x)) = J xh by Example 14 = J f(x)g · J xf by the Chain Rule = (D 1g(f(x))) · (D 1f(x)) = (g0(f(x))) · (f0(x)) = (g0(f(x))f0(x)) This is the well-known 1-dimensional version of the chain rule. Aug 12, 2020 · Chain Rules for One or Two Independent Variables Recall that the chain rule for the derivative of a composite of two functions can be written in the form d dx(f(g(x))) = f′ (g(x))g′ (x). In this equation, both f(x) and g(x) are functions of one variable. chain rule. By doing all of these things at the same time, we are more likely to make errors, at least until we have a lot of experience. 1.1 Expanding notation into explicit sums and equations for each component In order to simplify a given calculation, it is often useful to write out the explicit formula for

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The chain rule from single variable calculus has a direct analogue in multivariable calculus, where the derivative of each function is replaced by its Jacobian matrix, and multiplication is replaced with matrix multiplication. This rule, which became final in September 2015, requires food facilities to have a food safety plan in place that includes an analysis of hazards and risk-based preventive controls to minimize or ...

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Matrix-Chain Multiplication • Let A be an n by m matrix, let B be an m by p matrix, then C = AB is an n by p matrix. • C = AB can be computed in O(nmp) time, using traditional matrix multiplication. • Suppose I want to compute A 1A 2A 3A 4. • Matrix Multiplication is associative, so I can do the multiplication in several diﬀerent ... The Multivariable Chain Rule Nikhil Srivastava February 11, 2015 The chain rule is a simple consequence of the fact that di erentiation produces the linear approximation to a function at a point, and that the derivative is the coe cient appearing in this linear approximation. Let’s see this for the single variable case rst.

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Feb 27, 2018 · Vector chain rule :-Vector chain rule for vectors of functions and a single parameter mirrors the single-variable chain rule. If y = f ( g ( x )) and x is a vector .

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May 31, 2018 · With the first chain rule written in this way we can think of \(\eqref{eq:eq1}\) as a formula for differentiating any function of \(x\) and \(y\) with respect to \(\theta \) provided we have \(x = r\cos \theta \) and \(y = r\sin \theta \). vec(A) The vector-version of the matrix A (see Sec. 10.2.2) sup Supremum of a set jjAjj Matrix norm (subscript if any denotes what norm) AT Transposed matrix A TThe inverse of the transposed and vice versa, A T = (A 1)T = (A ) . A Complex conjugated matrix AH Transposed and complex conjugated matrix (Hermitian) A B Hadamard (elementwise) product A Then, DF= 2xy3z43x2y2z44x2y3z3. exsin(yz) zexcos(yz) yexcos(yz) : 3. The Chain Rule Stating the Chain Rule in terms of the derivative matrices is strikingly similar to the well-known (f g)0(x) = f0(g(x)) g0(x). The main di erence is that we use matrix multiplication!

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Oct 01, 2010 · The chain rule for α-derivatives, on the other hand, is simple and straightforward. For example, suppose we have a scalar function of a square matrix X, φ (X) = log | X |. Then, d φ (X) = tr X − 1 d X = (vec X ′ − 1) ′ d vec X, and hence D φ (X) = (vec X ′ − 1) ′. Now we are told that in fact X depends on a vector θ. Then, it ... The chain rule is a rule for differentiating compositions of functions. In the following discussion and solutions the derivative of a function h(x) will be denoted by or h'(x) . Most problems are average. A few are somewhat challenging. The chain rule states formally that . However, we rarely use this formal approach when applying the chain ... The chain rule from single variable calculus has a direct analogue in multivariable calculus, where the derivative of each function is replaced by its Jacobian matrix, and multiplication is replaced with matrix multiplication.

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Aug 12, 2020 · Chain Rules for One or Two Independent Variables Recall that the chain rule for the derivative of a composite of two functions can be written in the form d dx(f(g(x))) = f′ (g(x))g′ (x). In this equation, both f(x) and g(x) are functions of one variable. The chain rule tells us how to find the derivative of a composite function. Brush up on your knowledge of composite functions, and learn how to apply the chain rule correctly. If you're seeing this message, it means we're having trouble loading external resources on our website. Using the notation of matrices of partial derivatives, we can rewrite the one-variable chain rule of equation (1) as (2) D h (t) = D f (g (t)) D g (t). Since matrix multiplication of 1 × 1 matrices is the same as scalar multiplication, this new equation is just equation (1) in disguised form. Chain rule examples: Exponential Functions. Differentiating using the chain rule usually involves a little intuition. The more times you apply the chain rule to different problems, the easier it becomes to recognize how to apply the rule. This section shows how to differentiate the function y = 3x + 1 2 using the chain rule. Using the notation of matrices of partial derivatives, we can rewrite the one-variable chain rule of equation (1) as (2) D h (t) = D f (g (t)) D g (t). Since matrix multiplication of 1 × 1 matrices is the same as scalar multiplication, this new equation is just equation (1) in disguised form.

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As you can probably imagine, the multivariable chain rule generalizes the chain rule from single variable calculus. The single variable chain rule tells you how to take the derivative of the composition of two functions: \dfrac {d} {dt}f (g (t)) = \dfrac {df} {dg} \dfrac {dg} {dt} = f' (g (t))g' (t) dtd f (g(t)) = dgdf

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MIT grad shows how to use the chain rule to find the derivative and WHEN to use it. To skip ahead: 1) For how to use the CHAIN RULE or "OUTSIDE-INSIDE rule",...

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A superscript T denotes the matrix transpose operation; for example, AT denotes the transpose of A. Similarly, if A has an inverse it will be denoted by A-1. The determinant of A will be denoted by either jAj or det(A). Similarly, the rank of a matrix A is denoted by rank(A). An identity matrix will be denoted by I, and 0 will denote a null matrix.

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Oct 01, 2010 · The chain rule for α-derivatives, on the other hand, is simple and straightforward. For example, suppose we have a scalar function of a square matrix X, φ (X) = log | X |. Then, d φ (X) = tr X − 1 d X = (vec X ′ − 1) ′ d vec X, and hence D φ (X) = (vec X ′ − 1) ′. Now we are told that in fact X depends on a vector θ. Then, it ...

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