Dynamic Exercises

© 2005-2019 Nagy Gábor Péter

Course: Applied Linear Algebra (CompSci)
Subject: Eigenvalues, eigenvector, matrix decompositions, linear programming
Date: February 17, 2020


1) Eigenvalues of a 2x2 matrix I.
Compute the eigenvalues of the matrix $B=\left[ \begin{array}{cc} 13 & -12 \\ 15 & -14 \end{array} \right]$.


2) Eigenvalues of a 2x2 matrix II.
Compute the eigenvalues of the matrix $B=\left[ \begin{array}{cc} 2 & 1 \\ -2 & 6 \end{array} \right]$.


3) Eigenvalues and eigenvectors of a 2x2 matrix.
Compute the eigenvalues and eigenvectors of the matrix $B=\left[ \begin{array}{cc} -4 & 0 \\ -3 & -1 \end{array} \right]$.


4) Diagonalization of a matrix.
Compute the matrices $A,D$ such that $D$ is diagonal and $B=ADA^{-1}$ holds for the matrix $B=\left[ \begin{array}{cc} 11 & -8 \\ 12 & -9 \end{array} \right]$.


5) Eigenvectors and eigenvalues of a triangular matrix.
Compute the eigenvectors and the corresponding eigenvalues of the matrix $B=\left[ \begin{array}{ccc} -6 & 3 & -5 \\ 0 & -2 & -1 \\ 0 & 0 & -1 \end{array} \right]$.


6) Eigenvectors and eigenvalues of a 3x3 matrix.
Compute the eigenvectors and the corresponding eigenvalues of the matrix $B=\left[ \begin{array}{ccc} 23 & 19 & 71 \\ -18 & -14 & -47 \\ -2 & -2 & -9 \end{array} \right]$.


7) QR decomposition.
Compute the QR decomposition of the matrix $A=\left[ \begin{array}{cc} 14 & -86 \\ -48 & 27 \end{array} \right]$.


8) Spectral decomposition.
Compute the spectral decomposition of the matrix $A=\left[ \begin{array}{cc} -117 & 60 \\ 60 & 2 \end{array} \right]$.


9) Square root of a positive semidefinite matrix.
Compute the square root of the positive semidefinite matrix $A=\left[ \begin{array}{cc} 25 & 60 \\ 60 & 144 \end{array} \right]$.


10) Singular values of a matrix.
Compute the singular values of the matrix $A=\left[ \begin{array}{ccc} 1 & -3 & 2 \\ 4 & 0 & 1 \end{array} \right]$.


11) 1-norm of a vector.
Compute the 1-norm of the vector $\mathbf v( -4, 2, 10 )$.


12) Infinite-norm of a vector.
Compute the $\infty$-norm of the vector $\mathbf v( 5, 10, -4 )$..


13) Line fitting in 1-norm.
Set the linar programming problem which describes the fitting line of the points $P_0(8,15)$, $P_1(15,8)$, $P_2(5,11)$, $P_3(9,5)$, $P_4(13,7)$ in 1-norm.


14) Line fitting in infinite-norm.
Set the linar programming problem which describes the fitting line of the points $P_0(12,11)$, $P_1(8,13)$, $P_2(6,13)$, $P_3(11,12)$, $P_4(15,13)$ in $\infty$-norm.