0
Given some matrix A, an eigenvector of A is a vector that, when acted on by A, will result in a scalar multiple of itself, i.e. Ax=[lambda]x, where lambda is a real scalar multiple, called an eigenvalue, and x is the eigenvector described.
To find x you will normally have to find lambda first, which means solving the "characteristic equation": det(A-[lambda]I)=0, where I is the identity matrix.
The derivation of the "characteristic equation" is as follows:
Rearrange the equation Ax=[lambda]x -> Ax-[lambda]x=0 -> (A-[lambda]I)x=0 and then use the property from linear algebra that says if (A-[lambda]x) has an inverse, then x=0. Since this is trivial, we must instead prove that (A-[lambda]x) does not have an inverse. Because the inverse of a matrix is equal to its transpose divided by its determinant, and because you can't divide by 0, a 0 valued determinant means that the inverse can't exist. This is why we must solve det(A-[lambda]I)=0 for lambda.
Once we have found lambda, we can put it in the equation Ax=[lambda]x, and it's then just a simple matter of solving the resulting linear equations.
What else can I help you with?
What is the echelon of a matrix?
The eigen values of a matirx are the values L such that Ax = Lxwhere A is a matrix, x is a vector, and L is a constant.The vector x is known as the eigenvector.
What are the entries of a matrix?
The eigen values of a matirx are the values L such that Ax = Lxwhere A is a matrix, x is a vector, and L is a constant.The vector x is known as the eigenvector.
What the difference between eigenvalue and eigenvector?
An eigenvalue is a scalar that indicates how much an eigenvector is stretched or compressed during a linear transformation represented by a matrix. In contrast, an eigenvector is a non-zero vector that remains in the same direction after the transformation, only scaled by the eigenvalue. Mathematically, for a square matrix (A), if (A\mathbf{v} = \lambda \mathbf{v}), then (\lambda) is the eigenvalue and (\mathbf{v}) is the corresponding eigenvector.
What are eigenvalues and eigenvectors?
An eigenvector is a vector which, when transformed by a given matrix, is merely multiplied by a scalar constant; its direction isn't changed. An eigenvalue, in this context, is the factor by which the eigenvector is multiplied when transformed.
How does AHP use eigenvalue and eigenvector?
how does ahp use eigen values and eigen vectors
What is the relationship between eigenvalue and mutual information?
I'm seeking the answer too. What's the meaning of the principal eigenvector of an MI matrix?
What is the significance of the unit eigenvector in the context of linear algebra and eigenvalues?
In linear algebra, the unit eigenvector is important because it represents a direction in which a linear transformation only stretches or shrinks, without changing direction. It is associated with an eigenvalue, which tells us the amount of stretching or shrinking that occurs in that direction. This concept is crucial for understanding how matrices behave and for solving systems of linear equations.
What is the echelon of a matrix?
The eigen values of a matirx are the values L such that Ax = Lxwhere A is a matrix, x is a vector, and L is a constant.The vector x is known as the eigenvector.
What are the eignvalues of a matrix?
The eigen values of a matirx are the values L such that Ax = Lxwhere A is a matrix, x is a vector, and L is a constant.The vector x is known as the eigenvector.
What are the entries of a matrix?
The eigen values of a matirx are the values L such that Ax = Lxwhere A is a matrix, x is a vector, and L is a constant.The vector x is known as the eigenvector.
What is equality of a matrix?
The eigen values of a matirx are the values L such that Ax = Lxwhere A is a matrix, x is a vector, and L is a constant.The vector x is known as the eigenvector.
What are eigen values and eigen vectors?
This is a complicated subject, which can't be explained in a few words. Read the Wikipedia article on "eigenvalue"; or better yet, read a book on linear algebra. Briefly, and quoting from the Wikipedia, "The eigenvectors of a square matrix are the non-zero vectors that, after being multiplied by the matrix, remain parallel to the original vector. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector is scaled when multiplied by the matrix."
What Albert Einstein invent in math?
He used EigenVector spaces in a unique way, that is still hard to this day to understand. See: http://www.einstein-online.info/spotlights/path_integrals
What is an eigenvalue?
If a linear transformation acts on a vector and the result is only a change in the vector's magnitude, not direction, that vector is called an eigenvector of that particular linear transformation, and the magnitude that the vector is changed by is called an eigenvalue of that eigenvector.Formulaically, this statement is expressed as Av=kv, where A is the linear transformation, vis the eigenvector, and k is the eigenvalue. Keep in mind that A is usually a matrix and k is a scalar multiple that must exist in the field of which is over the vector space in question.
