Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville

By Ian Goodfellow, Yoshua Bengio, Aaron Courville

The Deep studying textbook is a source meant to aid scholars and practitioners input the sphere of computing device studying often and deep studying particularly. the net model of the publication is now whole and may stay on hand on-line at no cost

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48) i The trace operator is useful for a variety of reasons. Some operations that are difficult to specify without resorting to summation notation can be specified using 46 CHAPTER 2. LINEAR ALGEBRA matrix products and the trace operator. 49) ||A||F = Tr(AA> ). Writing an expression in terms of the trace operator opens up opportunities to manipulate the expression using many useful identities. For example, the trace operator is invariant to the transpose operator: Tr(A) = Tr(A> ). 51) n n−1 Y Y (i) (n) Tr( F ) = Tr(F F (i) ).

The singular value decomposition (SVD) provides another way to factorize a matrix, into singular vectors and singular values. The SVD allows us to discover some of the same kind of information as the eigendecomposition. However, the SVD is 44 CHAPTER 2. LINEAR ALGEBRA more generally applicable. Every real matrix has a singular value decomposition, but the same is not true of the eigenvalue decomposition. For example, if a matrix is not square, the eigendecomposition is not defined, and we must use a singular value decomposition instead.

11 has at most one solution for each value of b. To do so, we need to ensure that the matrix has at most m columns. Otherwise there is more than one way of parametrizing each solution. Together, this means that the matrix must be square, that is, we require that m = n and that all of the columns must be linearly independent. A square matrix with linearly dependent columns is known as singular. If A is not square or is square but singular, it can still be possible to solve the 38 CHAPTER 2. LINEAR ALGEBRA equation.

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