Eigenvalues and Eigenvectors

1. Eigenvalues and Eigenvectors Computations And Geometry

2. Eigenvalues and Eigenvectors Let A be a matrix if there exists a scalar, and x a nonzero vector such that . We call an eigenvalue of the matrix A and x an eigenvector of the matrix A. is a Eigenvalue for A is a Eigenvector for A Numerical Interpretation of Eigenvalues In terms of matrix arithmetic eigenvalues turn matrix multiplication into scalar multiplication. Numerically knowing an eigenvalue tremendously lowers the number of operations required to get the result of a matrix multiplication. This optimizes the performance of certain algorithms that are used in the areas of computer graphics and engineering. The question that arises in this chapter is how can we determine what the eigenvalues and eigenvectors of a matrix are? What properties do eigenvalues of a matrix have? Eigenvalues and Singular Matrices Remember a square matrix A is singular if and only if there exists a nonzero solution x to the matrix equation . In order to apply this we subtract from both sides of the equation, but we can not immediately factor out x since A is a matrix and a scalar the operation is not defined. Multiply x by the identity matrix I, we get that is a eigenvalue for A when is singular, or the eigenvalues of A are the values that make the matrix singular. If and only if:

3. Eigenvectors and Null Spaces If the matrix is singular then all nonzero vectors x such that are the eigenvectors of A. All vectors x such that is the null space of A. This we call the eigenspace of the eigenvalue . Then, Remember a matrix is singular if and only if . This gives a method for how to find the eigenvalues of matrices. Finding Eigenvalues of Matrices To find the eigenvalues of a matrix we do the following: Form the matrix this amounts to subtracting down the main diagonal of the matrix. Solve the equation for . Treat as the variable and either factor this or use the quadratic equation if it can not be factored. Singular, if and only if, Finding Eigenvectors of Matrices After you have found each eigenvalue , to find the eigenvectors associated with an eigenvalue find a basis for since basis vectors are never zero. x a basis vector for:

4. Example Find the eigenvalues and associated eigenvectors for the matrix A given to the right. , which is singular if Solve this equation for by first multiplying it out then trying to factor it or else use the quadratic equation if it will not factor. From this we see that or . For the eigenvalue -3, Row reducing, we get: Vector form of solution: Check, Eigenvalue -3 with eigenvector . For the eigenvalue 2, Row reducing, we get: Vector form of solution: Check, Eigenvalue 2 with eigenvector .

5. Geometric Interpretation of Real Eigenvectors If is a linear transformation given by . If x is an eigenvector for M with eigenvalue then the result of applying T to x changes the length of x (maybe points in opposite direction ). Example Let be a reflection through the y-axis and M the matrix so that . Find the eigenvalues and eigenvectors for M. From the picture we get , singular if y x Eigenvalues: , Row reduces to: Vector solution: Eigenvector for -1 is: , Row reduces to: Vector solution: Eigenvector for 1 is: Vectors on the positive x-axis are sent to negative x-axis and visa versa. While vectors on the y-axis remain fixed.

6. Example Let be a projection onto the vector (i.e. . Find the matrix M such that and compute its eigenvalues and eigenvectors. Singular if, Or, , Row reduces to: Vector solution: Eigenvector for 0 is: Factoring this we get this gives eigenvalues and . , Row reduces to: Vector solution: Eigenvector for 0 is: For a projection, , 1 is an eigenvalue If v is orthogonal to w, , 0 is an eigenvalue

7. Example Let be a counterclockwise rotation of (i.e. ). Find the matrix M such that and compute its eigenvalues and eigenvectors. , Singular if, The equation has no real solutions, so this matrix has no real eigenvalues. In fact we can see geometrically that a rotation do not fix or reverse the direction of any vector in the plane. This matrix has no real eigenvalues. The eigenvalues for this matrix are and which are known as imaginary numbers. More advanced course in linear algebra give an interpretation to these values but for right now we will say this matrix has no real eigenvalues. Because the matrix has no eigenvalues it has no corresponding eigenvectors also.