Suppose A is an n×n symmetric positive definite matrix and A=B2. Given that the eigenvalues of A are 18.5 and 9.5, find the eigenvalues of B.

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BlackTom AI · Accepted Answer

We are given that A is an n×n symmetric positive definite matrix and A = B^2. The eigenvalues of A are 18.5 and 9.5. Our task is to infer the eigenvalues of B.

First, recall a key fact: if a matrix B is diagonalizable in a basis of eigenvectors that also diagonalizes A = B^2, then the eigenvalues of A are the squares of the eigenvalues of B. In particular, for each eigenpair (λ_B, v) of B, with B v = λ_B v, applying B^2 gives B^2 v = λ_B^2 v, so λ_A = λ_B^2 for the corresponding eigenvalue λ_A of A.

Since A is symmetric positive definite, all its eigenvalues are positive: here, 18.5 and 9.5. Therefore the eigenvalues of B, up to sign, should satisfy λ_B^2 ∈ {18.5, 9.5}, i.e. λ_B ∈ {±sqrt(18.5), ±sqrt(9.5)}.

Compute the square roots:
- sqrt(18.5) ≈ 4.301...
- sqrt(9.5) ≈ 3.082...

Thus, the eigenvalues of B are ±4.301... and ±3.082... depending on the choice of signs for each eigencomponent. If there is an additional constraint that B is symmetric positive semidefinite (a common convention when taking a square root of a positive definite matrix), we would select the nonnegative roots, giving eigenvalues approximately 4.301 and 3.082.

Note on the provided answer candidate: any numerical pair like 5.75 and 1.7 does not satisfy λ_B^2 = λ_A for either λ_A = 18.5 or λ_A = 9.5, since 5.75^2 = 33.0625 and 1.7^2 = 2.89, which are not equal to 18.5 or 9.5. This indicates a mismatch between the candidate values and the required relationship between eigenvalues of B and A.

Additionally, if B is not constrained to be symmetric or positive semidefinite, both positive and negative square roots are mathematically allowable for each eigenvalue, leading to the set {±sqrt(18.5), ±sqrt(9.5)} as possible eigenvalues of B. If further information specifies a particular sign convention, that would select the exact signs.

25S-STATS-102B-LEC-3 Quiz 1- Requires Respondus LockDown Browser

Suppose A is an n×n symmetric positive definite matrix and A=B2. Given that the eigenvalues of A are 18.5 and 9.5, find the eigenvalues of B.

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