eigenvalues

Consider the differential equation $\dot{w} = Aw$, with $w(t = 0) = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$. If $w(t) = e^t\vec{u}_x + e^{-2t}\vec{u}_y$ be the solution to the equati...

Consider a system where $x_1(t), x_2(t)$, and $x_3(t)$ are three internal state signals and $u(t)$ is the input signal. The differential equations governing the system are given by...

Consider the matrix $\begin{bmatrix}1 & k \\ 2 & 1\end{bmatrix}$, where $k$ is a positive real number. Which of the following vectors is/are eigenvector(s) of this matrix?

Let the sets of eigenvalues and eigenvectors of a matrix B be {λk | 1 ≤ k ≤ n} and {vk | 1 ≤ k ≤ n}, respectively. For any invertible matrix P, the sets of eigenvalues and eigenvec...

The state equation of a second order system is $\dot{x}(t) = Ax(t)$, $x(0)$ is the initial condition. Suppose $\lambda_1$ and $\lambda_2$ are two distinct eigenvalues of A and $v_1...

Let the sets of eigenvalues and eigenvectors of a matrix B be $$\{ {\lambda _k}|1 \le k \le n\} $$ and $$\{ {v_k}|1 \le k \le n\} $$, respectively. For any invertible matrix P, the...

The state equation of a second order system is $$x(t) = Ax(t),\,\,\,\,x(0)$$ is the initial condition. Suppose $$\lambda_1$$ and $$\lambda_2$$ are two distinct eigenvalues of A and...

Let $$\alpha$$, $$\beta$$ two non-zero real numbers and v 1 , v 2 be two non-zero real vectors of size 3 $$\times$$ 1. Suppose that v 1 and v 2 satisfy $$v_1^T{v_2} = 0$$, $$v_1^T{...

A real 2 $$\times$$ 2 non-singular matrix A with repeated eigen value is given as $$A = \left[ {\matrix{ x & { - 3.0} \cr {3.0} & {4.0} \cr } } \right]$$ where x is a real positive...

The number of distinct eigenvalues of the matrix $A = \begin{bmatrix} 2 & 2 & 3 & 3 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 3 & 3 \\ 0 & 0 & 0 & 2 \end{bmatrix}$ is equal to ________.

Let M be a real 4 $$ \times $$ 4 matrix. Consider the following statements : S1: M has 4 linearly independent eigenvectors. S2: M has 4 distinct eigenvalues. S3: M is non-singular...

Consider the $$5 \times 5$$ matrix $$A = \left[ {\matrix{ 1 & 2 & 3 & 4 & 5 \cr 5 & 1 & 2 & 3 & 4 \cr 4 & 5 & 1 & 2 & 3 \cr 3 & 4 & 5 & 1 & 2 \cr 2 & 3 & 4 & 5 & 1 \cr } } \right]$...

The value of $$x$$ for which the matrix $$A = \left[ {\matrix{ 3 & 2 & 4 \cr 9 & 7 & {13} \cr { - 6} & { - 4} & { - 9 + x} \cr } } \right]$$ has zero as an eigen value is _________...

Consider a $$2 \times 2$$ square matrix $$A = \left[ {\matrix{ \sigma & x \cr \omega & \sigma \cr } } \right]$$ Where $$x$$ is unknown. If the eigenvalues of the matrix $$A$$ are $...

The value of $$'x'$$ for which all the eigenvalues of the matrix given below are real is $$\left[ {\matrix{ {10} & {5 + j} & 4 \cr x & {20} & 2 \cr 4 & 2 & { - 10} \cr } } \right]$...

$$A$$ real $$\left( {4\,\, \times \,\,4} \right)$$ matrix $$A$$ satisfies the equation $${A^2} = {\rm I},$$ where $${\rm I}$$ is the $$\left( {4\,\, \times \,\,4} \right)$$ identit...

Consider the matrix $${J_6} = \left[ {\matrix{ 0 & 0 & 0 & 0 & 0 & 1 \cr 0 & 0 & 0 & 0 & 1 & 0 \cr 0 & 0 & 0 & 1 & 0 & 0 \cr 0 & 0 & 1 & 0 & 0 & 0 \cr 0 & 1 & 0 & 0 & 0 & 0 \cr 1 &...

Which one of the following statements is NOT true for a square matrix $$A$$?

The minimum eigenvalue of the following matrix is $$\left[ {\matrix{ 3 & 5 & 2 \cr 5 & {12} & 7 \cr 2 & 7 & 5 \cr } } \right]$$

The eigen values of a skew-symmetric matrix are

The eigen values of the following matrix $$\left[ {\matrix{ { - 1} & 3 & 5 \cr { - 3} & { - 1} & 6 \cr 0 & 0 & 3 \cr } } \right]$$ are

All the four entries of $$2$$ $$x$$ $$2$$ matrix $$P = \left[ {\matrix{ {{p_{11}}} & {{p_{12}}} \cr {{p_{21}}} & {{p_{22}}} \cr } } \right]$$ are non-zero and one of the eigen valu...

Consider a linear system whose state space Representation is $$\mathop x\limits^ \bullet \left( t \right) = AX\left( t \right).$$ If the initial state vector of the system is $$x\l...

The eigen values and the correspondinng eigen vectors of a $$2 \times 2$$ matrix are given by Eigen value $${\lambda _1} = 8$$ $${\lambda _2} = 4$$ Eigen vector $${V_1} = \left[ {\...

For the matrix $$\left[ {\matrix{ 4 & 2 \cr 2 & 4 \cr } } \right].$$ The eigen value corresponding to the eigen vector $$\left[ {\matrix{ {101} \cr {101} \cr } } \right]$$ is

A linear system is equivalently represented by two sets of state equations. $$\mathop x\limits^ \bullet = \,\,{\rm A}X\,\, + BU$$ and $$\mathop W\limits^ \bullet = \,\,CW\,\, + DU....

Given the matrix $$\left[ {\matrix{ { - 4} & 2 \cr 4 & 3 \cr } } \right],$$ the eigen vector is

If A = $$\left[ {\matrix{ { - 2} & 2 \cr 1 & { - 3} \cr } } \right],$$ then sin At is

A certain linear, time-invariant system has the state and output representation shown below: $$$\eqalign{ & \left[ {\matrix{ {\mathop {{x_1}}\limits^ \bullet } \cr {\mathop {{x_2}}...

The eigen values of the matrix $$\left[ {\matrix{ 2 & { - 1} & 0 & 0 \cr 0 & 3 & 0 & 0 \cr 0 & 0 & { - 2} & 0 \cr 0 & 0 & { - 1} & 4 \cr } } \right]$$ are

For the system described by the state equation $$$\mathop x\limits^ \bullet = \left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr {0.5} & 1 & 2 \cr } } \right]x + \left[ {\matrix{ 0 \cr 0...

The eigen values of the matrix $$A = \left[ {\matrix{ 0 & 1 \cr 1 & 0 \cr } } \right]$$ are

A linear second order single input continuous-time system is described by the following set of differential equations $$$\eqalign{ & \mathop {{x_1}}\limits^ \bullet \left( t \right...