How many eigenvalues can a matrix have

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Martin Golubitsky and Michael Dellnitz. The characteristic polynomial of the matrix is. An eigenvalue of is a root of the characteristic polynomial.

It is known as the characteristic polynomial of A. Thus l is an eigen value of A iff l is a zero or root of the characteristic polynomial f A x in F. The equation. Note that the coefficients of the characteristic polynomial are elements of field F under consideration.

If an eigenvalue l of A is known, the corresponding eigenvector s may be obtained by elementary row operations ero's performed on the matrix part of the augmented matrix [A- l I 0]. The algebraic multiplicity of an eigenvalue l of A is the highest k such that x- l k is a factor of f A x. This eigenspace is the same as N A- l I. The geometric multiplicity of the eigenvalue 0 of is 2, while that of the eigenvalue 0 of is 1.

The algebraic multiplicities of the eigenvalue 0 of both these matrices equal 2. A collection of eigenvalues of A in which an eigenvalue is repeated according to is algebraic multiplicity is called the spectrum of A, and is denoted by s A. Thus, e. The similarity relation therefore is an equivalence relation. It is easy to verify that the characteristic polynomial of a companion matrix of a monic polynomial q x is the monic poynomial q x itself e.

The matrix C -1 AC is said to be similar to A. The characteristic polynomial remains invariant under a similarity transform, i. The characteristic polynomial of each of the above companion matrices being the polynomial q x , to find the characteristic polynomial of A, Danilevsky's method applies a succession of similarity transforms to A to reduce it to a companion matrix form Q 4.

For the latter, the method could be separately applied to Z. X looks like:. Find the characteristic polynomials both by expanding the determinant xI-A and by the Danilevsky's method , eigenvalues, the associated algebraic and geometric multiplicities and the eigenspaces of the following matrices:. When is it possible to write every vector in F n as a linear combination of eigenvectors of a matrix A? Give an example of a matrix A for which it is possible.

Give another example for which it is not. W is non-singular and the columns of E are eigen-vectors of A show that. Show that similar matrices have the same eigenvalues, each of which has the same algebraic and the same geometric multiplicities.

Verify that similarity is an equivalence relation, i. Prove that the geometric multiplicity of an eigenvalue cannot exceed its algebraic multiplicity. Verify that the transformations mentioned in the Danilevsky's method indeed constitute similarity transforms.

Verify that the matrices Q 1 - Q 4 are similar and that their characteristic polynomials are equal to q x. Thus we have proved the following:. Theorem Diagonalization. A precise characterization of diagonalizable matrices involves the notion of the minimal polynomial of a matrix and will appear later.

For the present, to determine if A is diagonalizable one has to first determine the eigenvalues of A and then if some of them are multiple, to check if there exist a corresponding number of linearly independent eigenvectors.

If A is diagonalizable and its distinct eigenvalues are c 1 , c 2 , Find a matrix A with distinct eigenvalues c 1 , c 2 , Prove that eigenvectors corresponding to distinct eigenvalues of a matrix are linearly independent. Which of the following matrices is diagonalizable over R :.

Show that: a all of these are diagonalizable over C ; b all of these are diagonalizable over R ; and c only those of these are diagonalizable over Q that have 5 as an eigenvalue. If an invertible complex matrix A equals its inverse A -1 , prove that it has a full set of eigenvectors.

Hence there exist scalars c 0 , c 1 , c 2 , The Cayley-Hamilton theorem asserts that the familiar characteristic polynomial f A x itself kills annihilates A:. Theorem Cayley-Hamilton. Let C ij x denote its i, j -th cofactor, which for each pair i, j is a matrix polynomial of degree n-1, at most. With e j 's as the standard unit vectors, we therefore have. Hence we can write. Premultiplying both sides of these equations by I, A, A 2 , The polynomial p A x is known as the minimal polynomial of A.

In particular, the minimal polynomial divides the characteristic polynomial. Indeed, p A x f A x is a rephrasing of the Cayley-Hamilton theorem. If l is an eigenvalue of A, P l is an eigenvalue of P A. The interest in the existence and construction of possible similar matrices of a particular type is due to several problems for similar matrices being related: e.

Thus, if B has a relatively simpler structure, it may enable an easier solution of problems involving A. The case when B could be managed to be a triangular matrix is discussed below. Proof : Let l be any of the eigenvalues of A and let x be a corresponding eigen-vector. Then S is a non-singular matrix and.

If it has an inverse, its rank is 8. So it has 8 eigenvectors I think? It doesn't matter whether matrix is invertible or not. Although if a matrix is invertible then it means it is full rank i. This matrix has only one linearly independent eigen vector. Add a comment. Active Oldest Votes. For any of this, it doesn't matter whether or not the eigenvalues are non-zero. So all vectors other than the basis vectors would become linearly dependent.

If there are eigenvalues, there are eigenvectors. Also, splitting field of the matrix. For example if the rotation is by 90 degrees then you are right there are no real eigen values but there are two complex eigen values. In fact ,practically complex numbers as eigen value is not of much use but they do exist atleast mathematically.

Show 2 more comments. Well, to be more precise: It depends on the underlying field Stefan Perko Stefan Perko Take e.