Linear Algebra : Eigenvalues and Eigenvectors

Study concepts, example questions & explanations for Linear Algebra

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Example Questions

Example Question #61 : Eigenvalues And Eigenvectors

Consider the diagonally dominant matrix

True or false: The Gershgorin discs of  are mutually disjoint.

Possible Answers:

False

True

Correct answer:

False

Explanation:

A Gershgorin discs (or Gershgorin circle) for a diagonally dominant matrix is a circle in the complex plane with one of the diagonal entries as a center. The radius of that particular Gershgorin disc is the sum of the absolute values in the same row as this diagonal element. 

Examining the matrix, we see that the radii of the circles corresponding to each diagonal element are as follows:

Each of the Gershgorin discs comprises all elements in the complex plane within 3 units of the diagonal element. Noting that 22 is 2 units away from both 20 and 24, it is immediately proved that at least two of these discs share an element and, consequently, are not disjoint. 

Example Question #62 : Eigenvalues And Eigenvectors

 is a nonsingular real matrix with four eigenvalues: .

True or false:  must have these same four eigenvalues. 

Possible Answers:

True

False

Correct answer:

True

Explanation:

One property of eigenvalues is that if  is nonsingular, the set of eigenvalues of  is exactly the set of reciprocals of eigenvalues of . The eigenvalues of  are , so the eigenvalues of these numbers are the reciprocals of these - in order, . This is the same set.

Example Question #62 : Eigenvalues And Eigenvectors

 where  and  is a diagonally dominant matrix. Note that every nondiagonal element is the same value

Give a necessary and sufficient condition for the four Gershgorin discs of  to be mutually disjoint. 

Possible Answers:

Correct answer:

Explanation:

A Gershgorin disc (or circle) for a diagonally dominant matrix is a closed circle in the complex plane with one of the diagonal entries as a center. The radius of that particular Gershgorin circle is the sum of the absolute values in the same row as this diagonal element. 

The four Gershgorin circles of  have their centers at , 13, 22, and 30 on the complex plane (and, specifically, on the real axis). The radius of each of the circles is ; for the circles to be mutually disjoint, the distance between their centers must be greater than .

The least distance between any two diagonal elements is , so

,

or

.

 is a necessary and sufficient condition for no two of the Gershgorin discs to intersect. 

Example Question #62 : Eigenvalues And Eigenvectors

A  matrix  has  as its set of eigenvalues.

Calculate .

Possible Answers:

Correct answer:

Explanation:

The determinant of a matrix  is equal to the product of its eigenvalues, so 

Example Question #65 : Eigenvalues And Eigenvectors

A  matrix  has  as its set of eigenvalues.

Give the trace of .

Possible Answers:

Correct answer:

Explanation:

The trace of a matrix is equal to the sum of its eigenvalues, so 

Example Question #63 : Eigenvalues And Eigenvectors

The characteristic equation of a matrix  is 

Give the trace of .

Possible Answers:

Correct answer:

Explanation:

The trace of a matrix is equal to the sum of the eigenvalues of the matrix. Since the characteristic equation has degree five, the matrix has five (not necessarily distinct) eigenvalues. If  are the eigenvalues of the matrix, then its characteristic equation is

When expanded, the coefficient of its degree-four term is. The degree-four term in the characteristic polynomial is "missing", so the implied coefficient is 0. 

 

It follows that the sum of the eigenvalues is 0, and that, consequently, .

Example Question #64 : Eigenvalues And Eigenvectors

The characteristic equation of a matrix  is 

Give the determinant of .

Possible Answers:

Correct answer:

Explanation:

The determinant of a matrix is equal to the product of the eigenvalues of the matrix. Since the characteristic equation has degree five, the matrix has five (not necessarily distinct) eigenvalues. If  are the eigenvalues of the matrix, then its characteristic equation is

When expanded, this polynomial has constant term . It follows that the product of the eigenvalues is , and that, consequently, .

Example Question #65 : Eigenvalues And Eigenvectors

The characteristic equation of a matrix  is 

Which statement must be true?

Possible Answers:

None of the other statements need be true.

Correct answer:

Explanation:

The determinant of a matrix is equal to the product of the eigenvalues of the matrix. Since the characteristic equation has degree four, the matrix has four (not necessarily distinct) eigenvalues. If  are the eigenvalues of the matrix, then its characteristic equation is

When expanded, this polynomial has constant term . It follows that the product of the eigenvalues is , and that, consequently, .

Example Question #66 : Eigenvalues And Eigenvectors

Consider the diagonally dominant matrix

Give the radius of the Gershgorin disc with center 20. 

Possible Answers:

Correct answer:

Explanation:

A Gershgorin disc for a diagonally dominant matrix is a circle in the complex plane with one of the diagonal entries as a center. The radius of the disc corresponding to that diagonal entry is the sum of the absolute values of the entries in the same row. 

Observe the elements on the second row, whose diagonal element is 20.

The sum of the absolute values other elements in that row - and the radius of the Gershgorin disc with its center at 20 - is

Example Question #70 : Eigenvalues And Eigenvectors

A matrix  has  as its set of eigenvalues.

Give the set of eigenvalues of .

Possible Answers:

Correct answer:

Explanation:

If is nonsingular, the set of eigenvalues of  is exactly the set of reciprocals of eigenvalues of . The eigenvalues of  are , so the eigenvalues of these numbers are the reciprocals of these. The reciprocal of  is 

.

Similarly, 

The set of eigenvalues of  is .

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