rue or false:a) Every linear operator in an n-dimensional vector space has n distinct eigen-values;b) If a matrix has one eigenvector, it has infinitely many eigenvectors;c) There exists a square real matrix with no real eigenvalues;d) There exists a square matrix with no (complex) eigenvectors;e) Similar matrices always have the same eigenvalues;f) Similar matrices always have the same eigenvectors;g) A non-zero sum of two eigenvectors of a matrix A is always an eigenvector;h) A non-zero sum of two eigenvectors of a matrix A corresponding to thesame eigenvalue λ is always an eigenvector
Question
rue or false:a) Every linear operator in an n-dimensional vector space has n distinct eigen-values;b) If a matrix has one eigenvector, it has infinitely many eigenvectors;c) There exists a square real matrix with no real eigenvalues;d) There exists a square matrix with no (complex) eigenvectors;e) Similar matrices always have the same eigenvalues;f) Similar matrices always have the same eigenvectors;g) A non-zero sum of two eigenvectors of a matrix A is always an eigenvector;h) A non-zero sum of two eigenvectors of a matrix A corresponding to thesame eigenvalue λ is always an eigenvector
Solution
a) False. Every linear operator in an n-dimensional vector space may not have n distinct eigenvalues. It is possible for a linear operator to have repeated eigenvalues.
b) False. If a matrix has one eigenvector, it does not necessarily have infinitely many eigenvectors. It is possible for a matrix to have only one eigenvector associated with a particular eigenvalue.
c) True. There exists a square real matrix with no real eigenvalues. This can happen when all the eigenvalues of the matrix are complex numbers.
d) False. Every square matrix has at least one eigenvector. It may not have any real eigenvectors, but it will always have complex eigenvectors.
e) True. Similar matrices always have the same eigenvalues. This is a property of similarity transformations.
f) False. Similar matrices do not always have the same eigenvectors. While they may share some eigenvectors, they can also have different eigenvectors.
g) False. A non-zero sum of two eigenvectors of a matrix A is not always an eigenvector. The sum of eigenvectors does not necessarily satisfy the eigenvalue equation.
h) True. A non-zero sum of two eigenvectors of a matrix A corresponding to the same eigenvalue λ is always an eigenvector. The sum of eigenvectors with the same eigenvalue will also satisfy the eigenvalue equation.
Similar Questions
Which of the following statement(s) is/are true with respect to eigenvalues andeigenvectors of a matrix?(A) The sum of the eigenvalues of a matrix equals the sum of the elements of theprincipal diagonal.(B) If is an eigenvalue of a matrix A, then1 is always an eigenvalue of itstranspose (AT).(C) If is an eigenvalue of an orthogonal matrix A, then1 is also an eigenvalue ofA.(D) If a matrix has n distinct eigenvalues, it also has n independent eigenvectors.Q.30 For studying wing vibrations, a wing of mass M and finite dimensions has beenidealized by assuming it to be supported using a linear spring of equivalentstiffness k and a torsional spring of equivalent stiffnessk as shown in the figure.The centre of gravity (CG) of the wing idealized as an airfoil is markedin the figure. The number of degree(s) of freedom for this idealized wing vibrationmodel is _________. (Answer in integer)
Let A be n × n matrix. True or false:a) AT has the same eigenvalues as A.b) AT has the same eigenvectors as A.c) If A is is diagonalizable, then so is AT
Mana dari pernyataan berikut yang benar mengenai vektor eigen?Which of the following statements is correct about eigenvectors?ASetiap matriks memiliki vektor eigenEvery matrix has eigenvectorsBSebuah matriks bisa memiliki lebih dari satu vektor eigenA matrix can have more than one eigenvectorCVektor eigen tidak memiliki nilai eigenThe eigen vector does not have eigen valuesDVektor eigen selalu memiliki norma satuThe eigenvector always has a unit norm
Every linear operator in an n-dimensional vector space has n distinct eigen-values
An n×n matrix A is diagonalizable if A has n linearly independent eigenvectors.A. TrueB. False
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