notes of matrices-1

notes of matrices-1 - Eigenvalues If the action of a matrix...

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Eigenvalues If the action of a matrix on a (nonzero) vector changes its magnitude but not its direction, then the vector is called an eigenvector of that matrix. A vector which is "flipped" to point in the opposite direction is also considered an eigenvector. Each eigenvector is, in effect, multiplied by a scalar, called the eigenvalue corresponding to that eigenvector. The eigenspace corresponding to one eigenvalue of a given matrix is the set of all eigenvectors of the matrix with that eigenvalue. Many kinds of mathematical objects can be treated as vectors: ordered pairs , functions , harmonic modes , quantum states , and frequencies are examples. In these cases, the concept of direction loses its ordinary meaning, and is given an abstract definition. Even so, if this abstract direction is unchanged by a given linear transformation, the prefix "eigen" is used, as in eigenfunction , eigenmode , eigenstate , and eigenfrequency . If a matrix is a diagonal matrix , then its eigenvalues are the numbers on the diagonal and its eigenvectors are basis vectors to which those numbers refer. For example, the matrix stretches every vector to three times its original length in the x-direction and shrinks every vector to half its original length in the y-direction. Eigenvectors corresponding to the eigenvalue 3 are any multiple of the basis vector [1, 0]; together they constitute the eigenspace corresponding to the eigenvalue 3. Eigenvectors corresponding to the eigenvalue 0.5 are any multiple of the basis vector [0, 1]; together they constitute the eigenspace corresponding to the eigenvalue 0.5. In contrast, any other vector, [2, 8] for example, will change direction. The angle [2, 8] makes with the x-axis has tangent 4, but after being transformed, [2, 8] is changed to [6, 4], and the angle that vector makes with the x-axis has tangent 2/3. Linear transformations of a vector space , such as rotation , reflection , stretching, compression, shear or any combination of these, may be visualized by the effect they produce on vectors . In other words, they are vector functions. More formally, in a vector space L , a vector function A is defined if for each vector x of L there corresponds a unique vector y = A ( x ) of L . For the sake of brevity, the parentheses around the vector on which the transformation is acting are often omitted. A vector function A is linear if it has the following two properties: Additivity : A ( x + y ) = A x + A y Homogeneity : A x ) = α A x where x and y are any two vectors of the vector space L and α is any scalar . [12] Such a function is variously called a linear transformation , linear operator , or linear endomorphism on the space L .
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Definition Given a linear transformation A , a non-zero vector x is defined to be an eigenvector of the transformation if it satisfies the eigenvalue equation for some scalar λ. In this situation, the scalar λ is called an eigenvalue of A corresponding to the eigenvector x . [13]
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notes of matrices-1 - Eigenvalues If the action of a matrix...

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