$$ \left.\begin{array}{l} ext { Given the covariance matrix } S=\left[\begin{array}{cc}4.0 & 15.0 \ 15.0 & 57.0\end{array} ight] ext { with eigenvalues } \lambda_{1}=60.95 ext { and } \lambda_{2}=0.05 ext { and } \ ext { eigenvectors } \mathbf{u}_{1}=\left[\begin{array}{c}-0.97 \ 0.25\end{array} ight] ext { and } \mathbf{u}_{2}=\left[\begin{array}{l}0.25 \ 0.97\end{array} ight] ext {, what is the diagonal matrix } D ext { such that } D=P^{T} S P \ D=[ ext { Ex: } 1.23 \ \square\end{array} ight] $$

Question

$$ \left.\begin{array}{l}	ext { Given the covariance matrix } S=\left[\begin{array}{cc}4.0 & 15.0 \ 15.0 & 57.0\end{array}ight] 	ext { with eigenvalues } \lambda_{1}=60.95 	ext { and } \lambda_{2}=0.05 	ext { and } \ 	ext { eigenvectors } \mathbf{u}_{1}=\left[\begin{array}{c}-0.97 \ 0.25\end{array}ight] 	ext { and } \mathbf{u}_{2}=\left[\begin{array}{l}0.25 \ 0.97\end{array}ight] 	ext {, what is the diagonal matrix } D 	ext { such that } D=P^{T} S P \ D=[	ext { Ex: } 1.23 \ \square\end{array}ight] $$

UpStudy AI · Accepted Answer

Given the covariance matrix

$$
S=\begin{pmatrix}
4.0 & 15.0 \
15.0 & 57.0
\end{pmatrix},
$$

its eigenvalues are given by

$$
\lambda_{1}=60.95, \quad \lambda_{2}=0.05,
$$

and the corresponding eigenvectors are

$$
\mathbf{u}_{1}=\begin{pmatrix}-0.97 \ 0.25\end{pmatrix}, \quad \mathbf{u}_{2}=\begin{pmatrix}0.25 \ 0.97\end{pmatrix}.
$$

If we form the matrix $$ P $$ by placing the eigenvectors as columns

$$
P = \begin{pmatrix}
-0.97 & 0.25 \
0.25  & 0.97
\end{pmatrix},
$$

then the diagonalization of $$ S $$ is written as

$$
S = PDP^{T},
$$

where $$ D $$ is the diagonal matrix containing the eigenvalues. Equivalently, by premultiplying by $$ P^{T} $$ and postmultiplying by $$ P $$ we have

$$
D = P^{T}SP.
$$

Since the eigenvectors are arranged in the same order as the eigenvalues (first column with $$ \lambda_1 $$ and second with $$ \lambda_2 $$), the diagonal matrix $$ D $$ is

$$
D = \begin{pmatrix}
60.95 & 0 \
0     & 0.05
\end{pmatrix}.
$$
The diagonal matrix $$ D $$ is $$ D = \begin{pmatrix} 60.95 & 0 \ 0 & 0.05 \end{pmatrix}. $$

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