- #1
Incand
- 334
- 47
As a step in a solution to another question our lecture notes claim that the matrix (a,b,c,d are real scalars).
\begin{bmatrix}
2a & b(1+d) \\
b(1+d)& 2dc \\
\end{bmatrix}
Is positive definite if the determinant is positive. Why? Since the matrix is symmetric it's positive definite if the it got positive (real) eigenvalues and ##det (A) = \prod \lambda_i## but two negative eigenvalues would give a positive determinant too.
Earlier in the text its given that the matrix
##
S =
\begin{bmatrix}
a & b \\
b& c \\
\end{bmatrix}
##
is positive definitive while ##d## is without any restrictions if that is somehow relevant.
\begin{bmatrix}
2a & b(1+d) \\
b(1+d)& 2dc \\
\end{bmatrix}
Is positive definite if the determinant is positive. Why? Since the matrix is symmetric it's positive definite if the it got positive (real) eigenvalues and ##det (A) = \prod \lambda_i## but two negative eigenvalues would give a positive determinant too.
Earlier in the text its given that the matrix
##
S =
\begin{bmatrix}
a & b \\
b& c \\
\end{bmatrix}
##
is positive definitive while ##d## is without any restrictions if that is somehow relevant.