Is Ax=wKx considered an eigenvalue problem in advanced linear algebra?

In summary, the conversation discusses the concept of eigenvalues and eigenvectors in linear algebra. It starts by explaining that in a previous course, eigenvalues are solutions to the system Ax=wx, where A is a square matrix, w is an eigenvalue, and x is an eigenvector. The system is solved by setting det(A-I*w)=0, where I is the identity matrix. In an advanced course, the equation system Ax=wKx is introduced, where A and K are square matrices, x is a vector, and w is a scalar. It is referred to as an eigenvalue-problem and can be solved by setting det(A-wK)=0. The conversation then raises a question about whether this is really an
  • #1
navalstudent
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From my Linear Algebra course I learned tha and eigenvalue w is an eigenvalue if it is a sollution to the system:

Ax=wx, where A= square matrix, w= eigenvalue, x= eigenvector. We solved the system by setting det(A-I*w)=0, I=identity matrix

Now in an advanced course I have come upon the equation system Ax=wKx, A= square matrix, x= vector, K= square matrix, w= scalar.

They say we can solve it by setting (A-wK)x=0, and they call this an eigenvalue-problem. And say we can solve it by settig det(A-wK)=0.

My question: Is this really an eigenvalue-problem? I looked in my book and on wikipedia, and there they both say that eigenvalue/vector prblems is Ax=wx, in my problem I have a matrix on both sides. Ax=wKx, so can w then be a eigenvalue?

I appreciate the help.
 
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  • #2
w is called a generalised eigenvalue of A and K and x is a generalised eigenvector. This form comes up frequently in differential equations. Click here to read more.
 

What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are concepts in linear algebra that are used to understand the properties of a linear transformation. Eigenvalues represent the scaling factor of the eigenvector when it is transformed by a linear transformation.

How are eigenvalues and eigenvectors calculated?

Eigenvalues and eigenvectors can be calculated by solving the characteristic equation of a square matrix. The characteristic equation is determined by subtracting the identity matrix multiplied by a scalar from the original matrix, and then finding the determinant of the resulting matrix.

What is the significance of eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are important in many areas of mathematics and science, including physics, engineering, and computer science. They can be used to understand the behavior of dynamical systems, analyze large datasets, and solve many types of differential equations.

Can a matrix have complex eigenvalues and eigenvectors?

Yes, a matrix can have complex eigenvalues and eigenvectors. This occurs when the matrix has complex entries or when the characteristic equation has complex solutions. Complex eigenvalues and eigenvectors can provide valuable insights into the behavior of a linear transformation.

What is the relationship between eigenvalues and determinants?

There is a direct relationship between eigenvalues and determinants. The determinant of a matrix is equal to the product of its eigenvalues. This means that the determinant can be used to find the eigenvalues of a matrix, and vice versa.

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