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Chapter 1: Q46E (page 1)

The product of the nsingular values of ann脳nmatrix A must be detA.

Short Answer

Expert verified

The given statement is TRUE.

Step by step solution

01

Step 1: Definition of the singular value decomposition

The diagonal entries of the matrix are singular values, which are arranged in ascending order.

Singular values are non-negative, which means it鈥檚 all positive real numbers.

02

Step 2: To Find TRUE or FALSE

Let蟽1,蟽2,鈰�,蟽nbe singular values ofA. Then,role="math" localid="1664186808738" 蟽12,蟽22,鈰�,蟽n2are eigenvalues ofATA.

Thus, 蟽12蟽22鈰�蟽n2=detATA=detA2, asAT=A.

Take square roots on both sides.

蟽1蟽2鈰�蟽n=detA

That is, the determinant of a matrix is equal to the product of the eigen values thus detAwill be the absolute value of the product of the eigen values which is in fact equal to the product of the singular values.

03

Final Answer

Thus, the given statement is TRUE.

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