Archetype K Archetype K
⬜ \(5\times 5\) nonsingular, diagonalizable, matrix
⬜ Summary Square matrix of size 5. Nonsingular. 3 distinct eigenvalues, 2 of multiplicity 2.
⬜ Definition A square matrix (Definition SQM). Notice how the following analysis parallels the analysis of the coefficient matrix of the linear systems in previous archetypes, yet there is no system discussed explicitly for this archetype.
\begin{equation*}
\begin{bmatrix}
10 & 18 & 24 & 24 & -12 \\
12 & -2 & -6 & 0 & -18 \\
-30 & -21 & -23 & -30 & 39 \\
27 & 30 & 36 & 37 & -30 \\
18 & 24 & 30 & 30 & -20
\end{bmatrix}
\end{equation*}
⬜ Row-Reduced Coefficient Matrix Row-equivalent matrix in reduced row-echelon form (Definition RREF).
\begin{equation*}
\begin{bmatrix}
\leading{1} & 0 & 0 & 0 & 0 \\
0 & \leading{1} & 0 & 0 & 0 \\
0 & 0 & \leading{1} & 0 & 0 \\
0 & 0 & 0 & \leading{1} & 0 \\
0 & 0 & 0 & 0 & \leading{1}
\end{bmatrix}
\end{equation*}
⬜ Coefficient Matrix Analysis Analysis of the reduced row-echelon form of the matrix (Definition RREF). For archetypes that begin as systems of equations, compare this analysis with the analysis for the coefficient matrices of the original system, and of the associated homogeneous system.
\begin{align*}
r&=5&D&=\set{1,\,2,\,3,\,4,\,5}&F&=\set{\ }
\end{align*}
⬜ Nonsingular Matrix? Is the matrix nonsingular or singular? Nonsingular. Notice that the row-reduced version of the matrix is the identity matrix and apply Theorem NMRRI. At the same time, examine the sizes of the sets \(D\) and \(F\) from the analysis of the reduced row-echelon version of the matrix.
⬜ Null Space The null space of the matrix. The set of vectors used in the span construction is a linearly independent set of column vectors that spans the null space of the matrix (Theorem SSNS, Theorem BNS). Solve a homogeneous system with this matrix as the coefficient matrix and write the solutions in vector form (Theorem VFSLS) to see these vectors arise. Compare the entries of these vectors for indices in \(D\) versus entries for indices in \(F\text{.}\)
\begin{equation*}
\set{\ }
\end{equation*}
⬜ Column Space, Original Columns The column space of the matrix, expressed as the span of a set of linearly independent vectors that are also columns of the matrix. These columns have indices that form the set \(D\) above (Theorem BCS).
\begin{equation*}
\set{\colvector{10\\12\\-30\\27\\18},\,\colvector{18\\-2\\-21\\30\\24},\,\colvector{24\\-6\\-23\\36\\30},\,\colvector{24\\0\\-30\\37\\30},\,\colvector{-12\\-18\\39\\-30\\-20}}
\end{equation*}
⬜ Column Space, Extended Echelon Form The column space of the matrix, as it arises from the extended echelon form of the matrix. The matrix \(L\) is computed as described in Definition EEF. This is followed by the column space described as the span of a set of linearly independent vectors that equals the null space of \(L\text{,}\) computed as according to Theorem FS and Theorem BNS. When \(r=m\text{,}\) the matrix \(L\) has no rows and the column space is all of \(\complex{m}\text{.}\)
\begin{equation*}
L=\begin{bmatrix}\end{bmatrix}
\end{equation*}
\begin{equation*}
\set{
\colvector{1\\0\\0\\0\\0},\,
\colvector{0\\1\\0\\0\\0},\,
\colvector{0\\0\\1\\0\\0},\,
\colvector{0\\0\\0\\1\\0},\,
\colvector{0\\0\\0\\0\\1}
}
\end{equation*}
⬜ Column Space, Row Space of Transpose The column space of the matrix, expressed as the span of a set of linearly independent vectors. These vectors are computed by bringing the transpose of the matrix into reduced row-echelon form, tossing out the zero rows, and writing the remaining nonzero rows as column vectors. By Theorem CSRST and Theorem BRS, and in the style of Example CSROI, this yields a linearly independent set of vectors that span the column space.
\begin{equation*}
\set{\colvector{1\\0\\0\\0\\0},\,\colvector{0\\1\\0\\0\\0},\,\colvector{0\\0\\1\\0\\0},\,\colvector{0\\0\\0\\1\\0},\,\colvector{0\\0\\0\\0\\1}}
\end{equation*}
⬜ Row Space Row space of the matrix, expressed as a span of a set of linearly independent vectors, obtained from the nonzero rows of the row-equivalent matrix in reduced row-echelon form. (Theorem BRS)
\begin{equation*}
\set{\colvector{1\\0\\0\\0\\0},\,\colvector{0\\1\\0\\0\\0},\,\colvector{0\\0\\1\\0\\0},\,\colvector{0\\0\\0\\1\\0},\,\colvector{0\\0\\0\\0\\1}}
\end{equation*}
⬜ Inverse Matrix? The matrix is nonsingular, and by Theorem NI has an inverse (Definition MI). The inverse can be computed with the procedure in Theorem CINM.
\begin{equation*}
\begin{bmatrix}
1 & -\left( \frac{9} {4} \right) & -\left( \frac{3}{2} \right) & 3 & -6 \\
\frac{21} {2} & \frac{43}{4} & \frac{21}{2} & 9 & -9 \\
-15 & -\left( \frac{21}{2} \right) & -11 & -15 & \frac{39} {2} \\
9 & \frac{15} {4} & \frac{9} {2} & 10 & -15 \\
\frac{9} {2} & \frac{3}{4} & \frac{3}{2} & 6 & -\left( \frac{19}{2} \right)
\end{bmatrix}
\end{equation*}
⬜ Subspace Dimensions Subspace dimensions associated with the matrix (Definition ROM, Definition NOM). Verify Theorem RPNC.
\begin{align*}
\text{rank}&=5&\text{nullity}&=0&\text{columns}&=5
\end{align*}
⬜ Determinant Value of the determinant of the matrix. The matrix is nonsingular so the determinant is nonzero (Theorem SMZD). Notice that zero is not an eigenvalue of the matrix (Theorem SMZE).
\begin{equation*}
\text{determinant}=16
\end{equation*}
⬜ Eigenvalues, Eigenspaces Eigenvalues, and bases for eigenspaces (Definition EEM, Definition EM). Compute a matrix-vector product (Definition MVP) for each eigenvector as an interesting check.
\begin{align*}
\eigensystem{K}{-2}{\colvector{2\\-2\\1\\0\\1},\,\colvector{-1\\2\\-2\\1\\0}}\\
\eigensystem{K}{1}{\colvector{4\\-10\\7\\0\\2},\,\colvector{-4\\18\\-17\\5\\0}}\\
\eigensystem{K}{4}{\colvector{1\\-1\\0\\1\\1}}\\
%& forces align environment
\end{align*}
⬜ Eigenvalue Multiplicities Geometric and algebraic multiplicities (Definition GME, Definition AME).
\begin{align*}
\geomult{K}{-2}&=2&\algmult{K}{-2}&=2\\
\geomult{K}{1}&=2&\algmult{K}{1}&=2\\
\geomult{K}{4}&=1&\algmult{K}{4}&=1
\end{align*}
⬜ Diagonalizable Diagonalizable (Definition DZM)?
Yes, full eigenspaces, Theorem DMFE.
⬜ Diagonalization The diagonalization (Theorem DC).
\begin{equation*}
\begin{bmatrix}-4&-3&-4&-6&7\\-7&-5&-6&-8&10\\
1&-1&-1&1&-3\\1&0&0&1&-2\\2&5&6&4&0
\end{bmatrix}\begin{bmatrix}
10 & 18 & 24 & 24 & -12 \\
12 & -2 & -6 & 0 & -18 \\
-30 & -21 & -23 & -30 & 39 \\
27 & 30 & 36 & 37 & -30 \\
18 & 24 & 30 & 30 & -20
\end{bmatrix}
\begin{bmatrix}2&-1&4&-4&1\\-2&2&-10&18&-1\\
1&-2&7&-17&0\\0&1&0&5&1\\1&0&2&0&1
\end{bmatrix}=\begin{bmatrix}-2&0&0&0&0\\0&-2&0&0&0\\
0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&4
\end{bmatrix}
\end{equation*}
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