Linear Algebra - Quadratic Forms and Positive Semidefinite Matrices

$\begin{align*}&\text{Use eigenvalues to determine if the following matrix is positive semi-definite:}\begin{bmatrix}12&-5\-5&9\end{bmatrix}\end{align*}$

$\text{The matrix is positive semi-definite.}$

$\text{The matrix is not positive semi-definite.}$

Explanation

$\begin{align*}&\text{The matrix given in the problem is symmetric; that is to say,}\&\text{it is identical to its transpose. If a matrix is symmetric,}\&\text{then it is also positive semi-definite if all of its eigenvalues}\&\text{are non-negative. Computing these eigenvalues:}\&\text{Eigenvalues of a matrix A, usually denoted as }\lambda\text{, are values which satisfy}\&det(A-\lambda I)\text{, where I is the identity matrix. For our matrix }A=\begin{bmatrix}12&-5\-5&9\end{bmatrix}\&\text{We can define a matrix of the form }\begin{bmatrix}12 - \lambda&-5\-5&9 - \lambda\end{bmatrix}\&\text{For a square matrix with dimensions of }2\&det\begin{vmatrix} a&b \ c&d \end{vmatrix}=ad-bc\&\lambda ^{2} - 21\lambda + 83\&\lambda_{1}=5.28;\lambda_{2}=15.72\&\text{The matrix is positive semi-definite.}\end{align*}$

$\begin{align*}&\text{See with eigenvalues if the following matrix is positive semi-definite:}\begin{bmatrix}-9&-15&8\-15&12&6\8&6&6\end{bmatrix}\end{align*}$

$\text{The matrix is not positive semi-definite.}$

$\text{The matrix is positive semi-definite.}$

Explanation

$\begin{align*}&\text{Observation of the matrix given will reveal that it is symmetric.}\&\text{In other words, it is equal to its transpose. If a matrix is}\&\text{symmetric, then it is also positive semi-definite if all of}\&\text{its eigenvalues are non-negative. So then, let us compute these}\&\text{eigenvalues:}\&\text{Eigenvalues of a matrix A, usually denoted as }\lambda\text{, are values which satisfy}\&det(A-\lambda I)\text{, where I is the identity matrix. For our matrix }A=\begin{bmatrix}-9&-15&8\-15&12&6\8&6&6\end{bmatrix}\&\text{We can define a matrix of the form }\begin{bmatrix}- \lambda - 9&-15&8\-15&12 - \lambda&6\8&6&6 - \lambda\end{bmatrix}\&\text{For a square matrix with dimensions of }3\&det\begin{vmatrix} a&b&c \ d&e&f\g&h&i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)\&(-1)\cdot (\lambda ^{3} - 9\lambda ^{2} - 415\lambda + 3882)\&\lambda_{1}=-20.49;\lambda_{2}=9.45;\lambda_{3}=20.04\&\text{The matrix is not positive semi-definite.}\end{align*}$

$\begin{align*}&\text{Decide whether or not the matrix }\begin{bmatrix}15&-11&8\-11&20&-1\8&-1&10\end{bmatrix}\&\text{is positive semi-definite using eigenvalues.}\end{align*}$

$\text{The matrix is positive semi-definite.}$

$\text{The matrix is not positive semi-definite.}$

Explanation

$\begin{align*}&\text{The matrix given in the problem is symmetric; that is to say,}\&\text{it is identical to its transpose. If a matrix is symmetric,}\&\text{then it is also positive semi-definite if all of its eigenvalues}\&\text{are non-negative. Computing these eigenvalues:}\&\text{Eigenvalues of a matrix A, usually denoted as }\lambda\text{, are values which satisfy}\&det(A-\lambda I)=0\text{, where I is the identity matrix. For our matrix }A=\begin{bmatrix}15&-11&8\-11&20&-1\8&-1&10\end{bmatrix}\&\text{We can define a matrix of the form }\begin{bmatrix}15 - \lambda&-11&8\-11&20 - \lambda&-1\8&-1&10 - \lambda\end{bmatrix}\&\text{For a square matrix with dimensions of }3\&det\begin{vmatrix} a&b&c \ d&e&f\g&h&i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)\&(-1)\cdot (464\lambda - 45\lambda ^{2} + \lambda ^{3} - 671)=0\&\lambda_{1}=1.72;\lambda_{2}=12.76;\lambda_{3}=30.51\&\text{The matrix is positive semi-definite.}\end{align*}$

$D = \begin{bmatrix} 4 & \log_{3} 4 & \log_{3}4 & \log_{3}4 \ \log_{5}6 & 4 & \log_{5}6& \log_{5} 6 \ \log_{7} 8 & \log_{7}8 & 4 & \log_{7}8 \ \log_{9} 10 & \log_{9}10 & \log_{9}10 & 4 \end{bmatrix}$

True or false: is an example of a diagonally dominant matrix.

True

False

Explanation

A real matrix is a diagonally dominant matrix if and only if, in each row, the absolute value of the diagonal element is greater than or equal to the sum of the absolute values of the other elements in that row. Below is with the diagonal elements in red:

$D = \begin{bmatrix}{\color{Red} \textbf{4} }& \log_{3} 4 & \log_{3}4 & \log_{3}4 \ \log_{5}6 & {\color{Red} \textbf{4} } & \log_{5}6& \log_{5} 6 \ \log_{7} 8 & \log_{7}8 & {\color{Red} \textbf{4} } & \log_{7}8 \ \log_{9} 10 & \log_{9}10 & \log_{9}10 & {\color{Red} \textbf{4} } \end{bmatrix}$

For each row, compare the absolute value of the diagonal element to the sum of the absolute values of the other elements. Note that each nondiagonal element is a logarithm, with base greater than 1, of a number greater than 1, so each element is positive; also, to calculate each nondiagonal sum, the sum rule for logarithms will be employed:

Row 1:

$\log_{3}4 + \log_{3}4 + \log_{3}4 = \log_{3} (4 \cdot 4 \cdot 4 )= \log_{3} 64$

$4 = \log_{3}(3^{4})= \log_{3} 81$

Therefore,

$|4 |>|\log_{3}4| +|\log_{3}4| +|\log_{3}4|$ .

Row 2:

$\log_{5}6 +\log_{5}6 +\log_{5}6 = \log_{5} (6 \cdot 6 \cdot 6 )= \log_{5} 216$

$4 = \log_{5}(5^{4}) =\log_{5}625$

Therefore,

$|4 |>|\log_{5}6| +|\log_{5}6| +|\log_{5}6|$ .

Row 3:

$\log_{7} 8 + \log_{7} 8+ \log_{7} 8 = \log_{7}( 8 \cdot 8 \cdot 8 )= \log_{7} 512$

$4 = \log_{7}(7^{4})= \log_{5}2,401$

Therefore,

$|4 |>|\log_{7} 8 | + |\log_{7} 8 | + |\log_{7} 8 |$ .

Row 4:

$\log_{9}10+ \log_{9}10 + \log_{9}10 = \log_{9}( 10 \cdot 10 \cdot 10 )= \log_{9}1,000$

$4 = \log_{9}(9^{4}) =\log_{9} 6,561$

Therefore,

$|4 |>| \log_{9}10| + | \log_{9}10| + | \log_{9}10 |$ .

In each row, the absolute value of the diagonal element is greater than or equal to the sum of the absolute values of the other elements. is diagonally dominant.

$\begin{align*}&\text{Decide whether or not the matrix }\begin{bmatrix}17&-5&9\-5&18&-5\9&-5&18\end{bmatrix}\&\text{is positive semi-definite using eigenvalues.}\end{align*}$

$\text{The matrix is positive semi-definite.}$

$\text{The matrix is not positive semi-definite.}$

Explanation

$\begin{align*}&\text{The matrix given in the problem is symmetric; that is to say,}\&\text{it is identical to its transpose. If a matrix is symmetric,}\&\text{then it is also positive semi-definite if all of its eigenvalues}\&\text{are non-negative. Computing these eigenvalues:}\&\text{Eigenvalues of a matrix A, usually denoted as }\lambda\text{, are values which satisfy}\&det(A-\lambda I)=0\text{, where I is the identity matrix. For our matrix }A=\begin{bmatrix}17&-5&9\-5&18&-5\9&-5&18\end{bmatrix}\&\text{We can define a matrix of the form }\begin{bmatrix}17 - \lambda&-5&9\-5&18 - \lambda&-5\9&-5&18 - \lambda\end{bmatrix}\&\text{For a square matrix with dimensions of }3\&det\begin{vmatrix} a&b&c \ d&e&f\g&h&i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)\&(-1)\cdot (805\lambda - 53\lambda ^{2} + \lambda ^{3} - 3625)=0\&\lambda_{1}=8.48;\lambda_{2}=14.01;\lambda_{3}=30.51\&\text{The matrix is positive semi-definite.}\end{align*}$

$\begin{align*}&\text{Use eigenvalues to determine if the following matrix is positive semi-definite:}\begin{bmatrix}6&5&12\5&19&3\12&3&7\end{bmatrix}\end{align*}$

$\text{The matrix is not positive semi-definite.}$

$\text{The matrix is positive semi-definite.}$

Explanation

$\begin{align*}&\text{The matrix given in the problem is symmetric; that is to say,}\&\text{it is identical to its transpose. If a matrix is symmetric,}\&\text{then it is also positive semi-definite if all of its eigenvalues}\&\text{are non-negative. Computing these eigenvalues:}\&\text{Eigenvalues of a matrix A, usually denoted as }\lambda\text{, are values which satisfy}\&det(A-\lambda I)=0\text{, where I is the identity matrix. For our matrix }A=\begin{bmatrix}6&5&12\5&19&3\12&3&7\end{bmatrix}\&\text{We can define a matrix of the form }\begin{bmatrix}6 - \lambda&5&12\5&19 - \lambda&3\12&3&7 - \lambda\end{bmatrix}\&\text{For a square matrix with dimensions of }3\&det\begin{vmatrix} a&b&c \ d&e&f\g&h&i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)\&(-1)\cdot (111\lambda - 32\lambda ^{2} + \lambda ^{3} + 1807)=0\&\lambda_{1}=-5.61;\lambda_{2}=13.18;\lambda_{3}=24.43\&\text{The matrix is not positive semi-definite.}\end{align*}$

$\begin{align*}&\text{See with eigenvalues if the following matrix is positive semi-definite:}\begin{bmatrix}-14&2\2&-15\end{bmatrix}\end{align*}$

$\text{The matrix is not positive semi-definite.}$

$\text{The matrix is positive semi-definite.}$

Explanation

$\begin{align*}&\text{Observation of the matrix given will reveal that it is symmetric.}\&\text{In other words, it is equal to its transpose. If a matrix is}\&\text{symmetric, then it is also positive semi-definite if all of}\&\text{its eigenvalues are non-negative. So then, let us compute these}\&\text{eigenvalues:}\&\text{Eigenvalues of a matrix A, usually denoted as }\lambda\text{, are values which satisfy}\&det(A-\lambda I)=0\text{, where I is the identity matrix. For our matrix }A=\begin{bmatrix}-14&2\2&-15\end{bmatrix}\&\text{We can define a matrix of the form }\begin{bmatrix}- \lambda - 14&2\2&- \lambda - 15\end{bmatrix}\&\text{For a square matrix with dimensions of }2\&det\begin{vmatrix} a&b \ c&d \end{vmatrix}=ad-bc\&29\lambda + \lambda ^{2} + 206=0\&\lambda_{1}=-16.56;\lambda_{2}=-12.44\&\text{The matrix is not positive semi-definite.}\end{align*}$

Consider the diagonally dominant matrix

$A = \begin{bmatrix} 20 & 0.1 & 0.2 & -0.3 & 0 \ 0.2 & -15 & 0.4 & 0.1 & -0.3 \ 0.3 & -0.2 & 18 & -0.2 & 0.2 \ 0.2 & - 0.2 & 0.5 & 16 & 0.2 \ 0.4 & 0 & 0.1 & 0.8 & -40 \end{bmatrix}$

Of the five Gershgorin discs of , give the center of the one with the least radius.

Explanation

The Gershgorin discs of a diagonally dominant matrix have as their centers the diagonal elements of the matrix. The radius of a disc is the sum of the absolute values of the nondiagonal elements in the same row as the given diagonal element. Therefore, note the diagonal elements of :

$A = \begin{bmatrix} \textbf{20 }& 0.1 & 0.2 & -0.3 & 0 \ 0.2 & \textbf{-15}& 0.4 & 0.1 & -0.3 \ 0.3 & -0.2 & \textbf{18}& -0.2 & 0.2 \ 0.2 & - 0.2 & 0.5 & \textbf{16}& 0.2 \ 0.4 & 0 & 0.1 & 0.8 & \textbf{-40} \end{bmatrix}$

The sums of the absolute values of the nondiagonal elements corresponding to each diagonal element are:

$20:|0.1|+ |0.2|+|-0.3| + |0|= \textbf{{\color{Red} 0.6}}$

The smallest of the Gershgorin discs is the one with radius 0.6, with center 20.

$\begin{align*}&\text{Decide whether or not the matrix }\begin{bmatrix}15&-15\-15&20\end{bmatrix}\&\text{is positive semi-definite using eigenvalues.}\end{align*}$

$\text{The matrix is positive semi-definite.}$

$\text{The matrix is not positive semi-definite.}$

Explanation

$\begin{align*}&\text{The matrix given in the problem is symmetric; that is to say,}\&\text{it is identical to its transpose. If a matrix is symmetric,}\&\text{then it is also positive semi-definite if all of its eigenvalues}\&\text{are non-negative. Computing these eigenvalues:}\&\text{Eigenvalues of a matrix A, usually denoted as }\lambda\text{, are values which satisfy}\&det(A-\lambda I)=0\text{, where I is the identity matrix. For our matrix }A=\begin{bmatrix}15&-15\-15&20\end{bmatrix}\&\text{We can define a matrix of the form }\begin{bmatrix}15 - \lambda&-15\-15&20 - \lambda\end{bmatrix}\&\text{For a square matrix with dimensions of }2\&det\begin{vmatrix} a&b \ c&d \end{vmatrix}=ad-bc\&\lambda ^{2} - 35\lambda + 75=0\&\lambda_{1}=2.29;\lambda_{2}=32.71\&\text{The matrix is positive semi-definite.}\end{align*}$

$A = \begin{bmatrix} 20 & 0.1 & 0.2 & -0.3 & 0 \ 0.2 & -15 & 0.4 & 0.1 & -0.3 \ 0.3 & -0.2 & 18 & -0.2 & 0.2 \ 0.2 & - 0.2 & 0.5 & 16 & 0.2 \ 0.4 & 0 & 0.1 & 0.8 & -40 \end{bmatrix}$

True or false: is an example of a diagonally dominant matrix.

True

False

Explanation

A matrix is diagonally dominant if, in each row, the absolute value of the diagonal element is greater than or equal to the sum of the absolute values of the other elements. Compare the quantities in each row:

$A = \begin{bmatrix} 20 & 0.1 & 0.2 & -0.3 & 0 \ 0.2 & -15 & 0.4 & 0.1 & -0.3 \ 0.3 & -0.2 & 18 & -0.2 & 0.2 \ 0.2 & - 0.2 & 0.5 & 16 & 0.2 \ 0.4 & 0 & 0.1 & 0.8 & -40 \end{bmatrix}$