linsys - 18.03, R05 LINEAR SYSTEMS 1 Eigenvalues and...

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18.03, R05 LINEAR SYSTEMS 1 Eigenvalues and eigenvectors of matrices A = a 11 a 12 . . . a 1 n a 21 a 22 . . . a 2 n . . . . . . . . . a n 1 a n 2 . . . a nn 1. The trace of A is the sum of the elements on the diagonal, Tr A = a 11 + a 22 + . . . + a nn . 2. The determinant of A is computed by expanding along a row or a column and keep doing it until we reduce the computation to 2 × 2 determinants. For instance, expanding along the i -th row we get det A = a i 1 ( - 1) i +1 det m i 1 + a i 2 ( - 1) i +2 det m i 2 + . . . + a in ( - 1) i + n det m in , where m ij stands for the ( i, j ) minor of A , namely the ( n - 1) × ( n - 1) matrix obtained from A by erasing the i -th row and the j -th column. 3. A is invertible if and only if det A 6 = 0. In that case A - 1 = 1 det A m 11 - m 12 . . . ( - 1) 1+ n m 1 n - m 21 m 22 . . . ( - 1) 2+ n m 2 n . . . . . . . . . ( - 1) n +1 m n 1 ( - 1) n +2 m n 2 . . . m nn T , where [ ] T stands for the transpose of a matrix and the m ij ’s denote minors like above. 4. The characteristic polynomial of A is the degree n polynomial p A ( λ ) = det( A - λI n ) . 5. The eigenvalues of A are the roots of p A ( λ ) = 0. Counting multiplicities there are n of them. 6. An eigenvector corresponding to the eigenvalue λ of the matrix A is a nonzero vector v such that A v = λ v , or equivalently, ( A - λI n ) v = 0 . 7. Assume λ is a repeated eigenvalue of A with multiplicity m . It is called complete if there are m linearly independent eigenvectors corresponding to it and defective otherwise. 8. If λ is a defective eigenvalue and v is a corresponding eigenvector, a generalized eigenvector is a vector u such that ( A - λI ) u = v . 1
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18.03, R05 2 Matrix exponentials Definition The exponential of a square matrix A is e A = X k =0 1 k ! A k . Note: If B in another n × n matrix, then e A + B = e A e B if and only if AB = BA . If A has eigenvalues λ 1 , . . . , λ n , the eigenvalues of its exponential e A are e λ 1 , . . . , e λ n . How to compute Probably the easiest way is to make a system x 0 = A x and find one of fundamental matrices F ( t ). Then e A = F (1) F (0) - 1 . There is one other trick that might help with computations, namely if all the elements on the main diagonal are equal to r , write A = rI + B . Then B to some power gives the zero matrix and its exponential is easy to compute being a finite sum and e A = e r e B . D
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linsys - 18.03, R05 LINEAR SYSTEMS 1 Eigenvalues and...

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