For a real n×n matrix A: I:if A is symmetric,there exists c≥0 for which A+cI is symmetric positive definite. II:if A is symmetric positive definite,there exists symmetric positive definite B with A=B². Which hold?
A symmetric matrix has real eigenvalues. Choose c>max(0,−λ_min) to make all shifted eigenvalues positive,proving I. For II,spectral decomposition A=Q diag(λ_i)Qᵀ with λ_i>0 gives B=Q diag(√λ_i)Qᵀ. Both hold. **C**.
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