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Recent questions tagged eigen-values

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TIFR ECE 2015 | Question: 13
Let \[ A=\left(\begin{array}{ccc} 1 & 1+\varepsilon & 1 \\ 1+\varepsilon & 1 & 1+\varepsilon \\ 1 & 1+\varepsilon & 1 \end{array}\right) \] Then for $\varepsilon=10^{-6}, A$ has only negative eigenvalues only non-zero eigenvalues only positive eigenvalues one negative and one positive eigenvalue None of the above

admin asked in Linear Algebra Dec 15, 2022

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TIFR ECE 2014 | Question: 7
Let $A$ be an $n \times n$ real matrix. It is known that there are two distinct $n$-dimensional real column vectors $v_{1}, v_{2}$ such that $A v_{1}=A v_{2}$. Which of the following can we conclude about $A?$ All eigenvalues of $A$ are non-negative. $A$ is not full rank. $A$ is not the zero matrix. $\operatorname{det}(A) \neq 0$. None of the above.

admin asked in Linear Algebra Dec 14, 2022

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TIFR ECE 2013 | Question: 12
Let $A$ be a Hermitian matrix and let $I$ be the Identity matrix with same dimensions as $A$. Then for a scalar $\alpha>0, A+\alpha I$ has the same eigenvalues as of $A$ but different eigenvectors the same eigenvalues and eigenvectors as of ... those of $A$ and same eigenvectors as of $A$ eigenvalues and eigenvectors with no relation to those of $A$ None of the above

admin asked in Linear Algebra Dec 12, 2022

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TIFR ECE 2012 | Question: 16
Let $P$ be a $n \times n$ matrix such that $P^{k}=\mathbf{0}$, for some $k \in \mathbb{N}$ and where $\mathbf{0}$ is an all zeros matrix. Then at least how many eigenvalues of $P$ are zero $1$ $n-1$ $n$ $0$ None of the above

admin asked in Linear Algebra Dec 8, 2022

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TIFR ECE 2012 | Question: 17
Let $A=U \Lambda U^{\dagger}$ be a $n \times n$ matrix, where $U U^{\dagger}=I$. Which of the following statements is TRUE. The matrix $I+A$ has non-negative eigen values The matrix $I+A$ is symmetic $\operatorname{det}(I+A)=\operatorname{det}(I+\Lambda)$ $(a)$ and $(c)$ $(b)$ and $(c)$ $(a), (b)$ and $(c)$

admin asked in Linear Algebra Dec 8, 2022

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TIFR ECE 2017 | Question: 7
A circulant matrix is a square matrix whose each row is the preceding row rotated to the right by one element, e.g., the following is a $3 \times 3$ circulant matrix. \[\left(\begin{array}{lll} 1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1 \ ... $j=\sqrt{-1}$ A vector whose $k$-th element is $\sinh \left(\frac{2 \pi k}{n}\right)$ None of the above

admin asked in Linear Algebra Nov 29, 2022

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GATE ECE 2010 | Question: 1
The eigenvalues of a skew-symmetric matrix are always zero always pure imaginary either zero or pure imaginary always real

admin asked in Linear Algebra Sep 15, 2022

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GATE ECE 2021 | Question: 36
A real $2\times2$ non-singular matrix $A$ with repeated eigenvalue is given as $A=\begin{bmatrix} x & -3.0\\ 3.0 & 4.0 \end{bmatrix}$ where $x$ is a real positive number. The value of $x$ (rounded off to one decimal place) is ________________

Arjun asked in Linear Algebra Feb 20, 2021

by Arjun

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9

GATE ECE 2019 | Question: 17
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 ____________.

Arjun asked in Linear Algebra Feb 12, 2019

by Arjun

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10

GATE ECE 2016 Set 2 | Question: 1
The value of $x$ for which the matrix $A= \begin{bmatrix} 3& 2 &4 \\ 9& 7 & 13\\ -6&-4 &-9+x \end{bmatrix}$ has zero as an eigenvalue is ________

Milicevic3306 asked in Linear Algebra Mar 28, 2018

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11

GATE ECE 2015 Set 2 | Question: 2
The value of $x$ for which all the eigen-values of the matrix given below are real is $\begin{bmatrix} 10&5+j &4 \\ x&20 &2 \\4 &2 &-10 \end{bmatrix}$ $5+j$ $5-j$ $1-5j$ $1+5j$

Milicevic3306 asked in Linear Algebra Mar 28, 2018

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12

GATE ECE 2015 Set 1 | Question: 5
The value of $p$ such that the vector $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ is an eigenvector of the matrix $\begin{bmatrix} 4 & 1 & 2 \\ p & 2 & 1 \\ 14 & -4 & 10 \end{bmatrix}$ is _________.

Milicevic3306 asked in Linear Algebra Mar 28, 2018

by Milicevic3306

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13

GATE ECE 2014 Set 3 | Question: 27
Which one of the following statements is NOT true for a square matrix $A$? If $A$ is upper triangular, the eigenvalues of $A$ are the diagonal elements of it If $A$ is real symmetric, the eigenvalues of $A$ are always real and positive If $A$ ... $A$ are positive, all the eigenvalues of $A$ are also positive

Milicevic3306 asked in Linear Algebra Mar 26, 2018

by Milicevic3306

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14

GATE ECE 2014 Set 1 | Question: 4
A real $(4 \times 4)$ matrix $A$ satisfies the equation $A^{2} = I$, where $I$ is the $(4 \times 4)$ identity matrix. The positive eigen value of $A$ is ______.

Milicevic3306 asked in Linear Algebra Mar 26, 2018

by Milicevic3306

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15

GATE ECE 2013 | Question: 19
The minimum eigenvalue of the following matrix is $\begin{bmatrix} 3& 5& 2\\5 &12 &7 \\2 &7 & 5\end{bmatrix}$ $0$ $1$ $2$ $3$

Milicevic3306 asked in Linear Algebra Mar 26, 2018

by Milicevic3306

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16

GATE ECE 2018 | Question: 11
Let $\text{M}$ 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 (invertible). Which one among the following is TRUE? $S1$ implies $S2$ $S1$ implies $S3$ $S2$ implies $S1$ $S3$ implies $S2$

gatecse asked in Linear Algebra Feb 19, 2018

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17

GATE ECE 2017 Set 1 | Question: 1
Consider the 5 $\times$ 5 matrix $\begin{bmatrix} 1&2&3&4&5\\ 5&1 &2& 3 &4\\ 4&5&1&2&3\\ 3&4&5&1&2\\ 2&3&4&5&1 \end{bmatrix}$ It is given that A has only one real eigenvalue. Then the real eigenvalue of A is $-2.5$ $0$ $15$ $25$

admin asked in Linear Algebra Nov 17, 2017

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