Question
Check which of the following matrices is positive definite and which is positive semi-definite:
Also, find the square root of the positive definite matrix.
Answer :
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To check whether a matrix is positive definite (PD) or positive semi-definite (PSD), we can use leading principal minors: * A symmetric matrix is positive definite if all leading principal minors are positive. * A symmetric matrix is positive semi-definite if all eigenvalues are non-negative or all principal minors are non-negative. --- Given matrices: [ A = \begin{bmatrix} 1 & 1 & 0 \ 1 & 2 & 1 \ 0 & 1 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 0 & 1 \ 0 & 2 & -1 \ 1 & -1 & 3 \end{bmatrix} ] --- ### Step 1: Check matrix (A) Check leading principal minors: 1. ( \Delta_1 = 1 > 0 ) ✅ 2. ( \Delta_2 = \begin{vmatrix} 1 & 1 \ 1 & 2 \end{vmatrix} = (1)(2) - (1)(1) = 1 > 0 ) ✅ 3. ( \Delta_3 = \det(A) = \begin{vmatrix} 1 & 1 & 0 \ 1 & 2 & 1 \ 0 & 1 & 1 \end{vmatrix} ) Expand along first row: [ \det(A) = 1 \cdot \begin{vmatrix} 2 & 1 \ 1 & 1 \end{vmatrix} - 1 \cdot
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