GATE MA concept:eigenvalues PYQs & Answers

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concept:eigenvalues

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Questions 1–10 of 10. Imported answers require independent review unless marked reviewed.

Topic practice 10 questionsPage 1 of 1

Let

be a normal matrix. Consider the following statements.

I. If all the eigenvalues of

are real, then

is Hermitian.

II. If all the eigenvalues of

have absolute value 1, then

is unitary.

Which one of the following is correct?

Your answerChoose one option.

Both I and II are TRUE

I is TRUE and II is FALSE

I is FALSE and II is TRUE

Both I and II are FALSE

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Let

be two non-zero, non-identity

-linear transformations. Assume

. Which of the following is/are TRUE?

Your answerSelect all correct options. You can choose more than one.

is necessarily invertible

and

are similar if

and

are similar if

has only 0 and 1 as eigenvalues

is necessarily diagonalizable

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Let

be an

-linear transformation such that 1 and 2 are the only eigenvalues of

. Suppose the dimensions of

and

are 1 and 2, respectively. Which of the following is/are possible (upper triangular) Jordan canonical form(s) of

Your answerSelect all correct options. You can choose more than one.

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Let

be the vector space of all polynomials of degree at most 2 over the real field

. Let

be the linear operator given by

Then the sum of the eigenvalues of

is __________.

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Suppose

is a finite dimensional non-zero vector space over

and

is a linear transformation such that

. Then which of the following statements is FALSE?

Your answerChoose one option.

The dimension of

is even

0 is the only eigenvalue of

Both 0 and 1 are eigenvalues of

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Let

be a

real symmetric matrix with eigenvalues

and

with the respective eigenvectors

and

Consider the following statements:

II. The vector

satisfies

III. For any

has a solution.

IV. The trace of the matrix

is 8.

(

denotes the transpose of the vector

)

Which of the above statements are TRUE?

Your answerChoose one option.

I, II and III only

I and II only

II and IV only

III and IV only

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Consider the matrix

with

, where

is the

identity matrix and

is the transpose of

. If

and

are two distinct eigenvalues of

, then

_________.

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Let

be a

matrix such that

, where

is the identity matrix. If

has two distinct eigenvalues and each eigenvalue has geometric multiplicity 3, then the total number of nonzero entries in the Jordan canonical form of

equals ______.

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Given two

matrices

and

with entries in

, consider the following statements: (P): If

and

have the same minimal polynomial, then

is similar to

. (Q): If

has

distinct eigenvalues, then there exists

such that

are linearly independent. Which of the above statements hold TRUE?

Your answerChoose one option.

Both P and Q

Only P

Only Q

Neither P nor Q

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Let

be a

matrix such that

for

and

, where

. Let

and

be the largest and the smallest eigenvalues of

, respectively. If

, then

equals ________.

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GATE MA concept:eigenvalues PYQs

How to practice concept:eigenvalues

Question 20

Question 51

Question 56

Question 52

Question 27

Question 36

Question 42

Question 2

Question 27

Question 28