matrix transpose of K. A definite matrix is positive or negative definite. A matrix
K is positive definite if z
T Kz > 0 and negative definite if z
T Kz < 0 for any test vector
z, where a test vector is any vector containing at least one non-zero element.
Example 2.2. Consider a vibration-related EVP (as in Example 2.1) having
real-valued, symmetric stiffness and mass matrix, and positive definite stiffness
matrix. By Theorem 2.1, the eigenvalues are all real-valued.
Many algebraic EVPs of applied mechanics possess eigenvectors that are
characterized by orthogonality. This property can be used for decoupling otherwise
coupled equations of motions, as illustrated in Sect. 1.3.6. We have:
Theorem 2.2 Orthogonality of eigenvectors. If the matrices K and L of the
EVP (2.1) are symmetric, then any two eigenvectors u i and u j are orthogona
l in the sense that u i
T
Ku j = u i
T Lu j = 0 for any i 6 ¼ j; i, j = 1, n.
2.2.3 Methods of Solution
There are numerous methods for calculating eigenvalues and eigenvectors of algebraic EVPs. The most simple relies on a direct calculation of the characteristic
polynomial, whereby roots k j (generally complex-valued) of the polynomial determinant equation |K − kL| = 0 are calculated, and in turn substituted back into the EVP
to yield the associated eigenvectors u j . This method is inefficient if K and L are large
matrices, and if more eigenvalues than the lowest few are in need. Then various
iterative methods become relevant, among these Jacobi’s method, inverse iteration
and (in particular powerful) subspace iteration (e.g., Bathe and Wilson 1976).
We illustrate here the method of inverse iteration. By this technique
vector-iterates u [k+1] are generated by solving the linear system of equations
Ku [k+1] = Lu [k] for k = 0, 1, …, starting with an arbitrary vector u [0] 6 ¼ 0. The
sequence u [1] , u [2] , … then converges towards the first eigenvector of the system.
For computational convenience a slightly modified algorithm is often used:
Inverse Iteration. For the EVP Ku = kLu:
1. Let u [0] = {1 1 ÁÁÁ 1}
T , k [0] = 1 and z [0] = Lu [0]
2. Iterate for k = 0, 1, …, until |k [k+1] − k [k] |/k [k] < e ( 1:
(a) Solve K ~
u [k+1] = z [k] to find ~
u [k+1]
(b) Compute ~ z [k+1] = L~ u [k+1] and r
2
½k þ 1 = ~
u
T
½k þ 1 ~ z ½k þ 1
(c) Compute k [k+1] = ~ u
T
½k þ 1 z k
½ =r
2
½k þ 1 and z [k+1] = ~ z [k+1] /r [k+1]
3. If iterations has converged at step k = p, then k 1 %k [ p+1] and
u 1 %~ u [p+1] /r [p+1]
2.2 The Algebraic EVP
51
K is positive definite if z
T Kz > 0 and negative definite if z
T Kz < 0 for any test vector
z, where a test vector is any vector containing at least one non-zero element.
Example 2.2. Consider a vibration-related EVP (as in Example 2.1) having
real-valued, symmetric stiffness and mass matrix, and positive definite stiffness
matrix. By Theorem 2.1, the eigenvalues are all real-valued.
Many algebraic EVPs of applied mechanics possess eigenvectors that are
characterized by orthogonality. This property can be used for decoupling otherwise
coupled equations of motions, as illustrated in Sect. 1.3.6. We have:
Theorem 2.2 Orthogonality of eigenvectors. If the matrices K and L of the
EVP (2.1) are symmetric, then any two eigenvectors u i and u j are orthogona
l in the sense that u i
T
Ku j = u i
T Lu j = 0 for any i 6 ¼ j; i, j = 1, n.
2.2.3 Methods of Solution
There are numerous methods for calculating eigenvalues and eigenvectors of algebraic EVPs. The most simple relies on a direct calculation of the characteristic
polynomial, whereby roots k j (generally complex-valued) of the polynomial determinant equation |K − kL| = 0 are calculated, and in turn substituted back into the EVP
to yield the associated eigenvectors u j . This method is inefficient if K and L are large
matrices, and if more eigenvalues than the lowest few are in need. Then various
iterative methods become relevant, among these Jacobi’s method, inverse iteration
and (in particular powerful) subspace iteration (e.g., Bathe and Wilson 1976).
We illustrate here the method of inverse iteration. By this technique
vector-iterates u [k+1] are generated by solving the linear system of equations
Ku [k+1] = Lu [k] for k = 0, 1, …, starting with an arbitrary vector u [0] 6 ¼ 0. The
sequence u [1] , u [2] , … then converges towards the first eigenvector of the system.
For computational convenience a slightly modified algorithm is often used:
Inverse Iteration. For the EVP Ku = kLu:
1. Let u [0] = {1 1 ÁÁÁ 1}
T , k [0] = 1 and z [0] = Lu [0]
2. Iterate for k = 0, 1, …, until |k [k+1] − k [k] |/k [k] < e ( 1:
(a) Solve K ~
u [k+1] = z [k] to find ~
u [k+1]
(b) Compute ~ z [k+1] = L~ u [k+1] and r
2
½k þ 1 = ~
u
T
½k þ 1 ~ z ½k þ 1
(c) Compute k [k+1] = ~ u
T
½k þ 1 z k
½ =r
2
½k þ 1 and z [k+1] = ~ z [k+1] /r [k+1]
3. If iterations has converged at step k = p, then k 1 %k [ p+1] and
u 1 %~ u [p+1] /r [p+1]
2.2 The Algebraic EVP
51