Determine the validity of each statement by selecting True or False. 1. If a 3×3 matrix has eigenvalues 𝜆=−3,−1,2, then the singular values of the matrix are 𝜎1=3,𝜎2=2,𝜎3=1. 2. The SVD of any 𝑚×𝑛 matrix 𝐴 always exists and is unique. 3. It is possible for a 3×4 matrix to have the singular values 𝜎1=4, 𝜎2=0, and 𝜎3=−1. 4. If 𝐴 has linearly independent columns and 𝐴 has the SVD 𝐴=𝑈Σ𝑉𝑇, then the columns of 𝑉 form an orthonormal basis for Col𝐴.
Question
Determine the validity of each statement by selecting True or False. 1. If a 3×3 matrix has eigenvalues 𝜆=−3,−1,2, then the singular values of the matrix are 𝜎1=3,𝜎2=2,𝜎3=1. 2. The SVD of any 𝑚×𝑛 matrix 𝐴 always exists and is unique. 3. It is possible for a 3×4 matrix to have the singular values 𝜎1=4, 𝜎2=0, and 𝜎3=−1. 4. If 𝐴 has linearly independent columns and 𝐴 has the SVD 𝐴=𝑈Σ𝑉𝑇, then the columns of 𝑉 form an orthonormal basis for Col𝐴.
Solution
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False. The singular values of a matrix are always non-negative. They are the square roots of the eigenvalues of the matrix A*A (where A is the original matrix). Therefore, they cannot be negative or complex if the matrix is real. In this case, the eigenvalues are -3, -1, and 2. The square roots of these values (which would be the singular values) would be complex for -3 and -1, which is not possible.
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True. The Singular Value Decomposition (SVD) of any m×n matrix A always exists. However, it is not always unique. If two or more singular values are the same, there can be more than one valid SVD for a matrix.
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False. As mentioned before, singular values are always non-negative. Therefore, a singular value of -1 is not possible.
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True. If A has linearly independent columns and A has the SVD A=UΣVT, then the columns of V do form an orthonormal basis for ColA. This is because V is an orthogonal matrix in the SVD, and the columns of an orthogonal matrix form an orthonormal basis.
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