Begin2.DVI

Bis said to be column equivalent to A. If B=P AQ, then Bis said to be equivalent to to the matrix A. All elementary matrices hav ...

left-hand side of the transformed augmented matrix. Consequently, the right-hand side of equation (10.27) becomes an equation, w ...

Eigenvalues and Eigenvectors Consider the operator box illustrated in the figure 10-5 where the input to the operator box is the ...

Solving this equation for the values of λgives the eigenvalues (λ 1 , λ 2 ,... , λ n)associated with the matrix A. Substituting ...

which implies x 1 =x 2. This gives the eigenvector x=col(x 1 , x 2 ) = col(x 2 , x 2 )where the component x 2 must be some nonze ...

given by col(1 , 0 ,0). Note that any nonzero constant times this vector is also an eigenvector. Substituting the eigenvalue λ= ...

The eigenvalues of Aare determined by the characteristic equation C(λ) = det (A−λI ) = |A−λI |= (−1)nλn+α 1 λn−^1 +··· +αn− 1 λ+ ...

3. If Ais a symmetric matrix and λiis an eigenvalue of multiplicity ri,then there are rilinearly independent eigenvectors. 4. An ...

λ 1 = 1, X 1 =col[ √ 3 ,1] and λ 2 = 2, X 2 =col[1,− √ 3]. Let Q= [X 1 , X 2 ] = [√ 3 1 1 − √ 3 ] = [ x 11 x 12 x 21 x 22 ] deno ...

Example 10-27. Find the eigenvalues and eigenvectors associated with the matrix A=   −1 4 6 − 6 1 4 0 2 −4 0 7 − 8 − 1 −2 0 0 ...

may or may not exist. The number Nof linearly independent eigenvectors associated with an eigenvalue λiis given by the formula N ...

Here Ais constant so that one can differentiate equation (10.37) with respect to t to obtain d dt ( eA t ) =A+A^2 t+A^3 t^2 2! + ...

Differentiate equation (10.40) with respect to tand show d dt sin(At) = A−A 3 t^2 2! +A 5 t^4 4! +···+ (−1) nA 2 n+1 t^2 n (2 n) ...

Example 10-29. The following is an example illustrating the Hamilton-Cayley theorem. Let A= [ 2 1 3 2 ] , then the characteristi ...

in any element of the matrix Adj(A−λI ).Observe that the equation Adj(A−λI )can be written in the form Adj(A−λI ) = B 1 λn−^1 +B ...

the summation produces the zero matrix. This establishes the Hamilton-Cayley theorem. Evaluation of Functions Let f(x)denote a s ...

which must exist between the functions f(x)and R(x).These equations are nindepen- dent relations one can use to solve for the un ...

and continuing in this manner, one finds the general form R(A) = f(A) = Ak=β 1 A+β 2 I for some constants β 1 and β 2 (i.e., f(A ...

Example 10-31. Given the matrix A= [ 2 − 1 −3 4 ] find the matrix function f(A) = eAt. Solution: Here f(x) = ext,and from the pr ...

Express f(A)as a linear combination of the matrices {I, A, A^2 }and write f(A) = R(A) = sin At =c 0 I+c 1 A+c 2 A^2 , where c 0 ...