Chapter 4 Determinants
1. Uses of Determinants
The determinant of
The determinant gives a formula for each pivot. From the formula
The determinant measures the dependence of
2. Properties of Determinants
- The determinant of the identity matrix is 1.
- The determinant changes sign when two rows are exchanged.
The determinant of every permutation matrix is
- The determinant depends linearly on the first row.
- If two rows of
are equal, then .
- Subtracting a multiple of one row from another row leaves the same determinant.
The usual elimination steps do not affect the determinant.
has a row of zeros, then .
is triangular, then is the product of the diagonal entries.
- The determinant of
is the product of times .
- The transpose of
has the same determinant as itself.
- LDU factorization.
3. Applications of Determinants
- Computation of
- Cramer's Rule.
Chapter 5 Eigenvalues and Eigenvectors
The eigenvalues are the most important feature of practically any dynamic system. Until now we have focused on the problem
(1) Solution of
This is a nonlinear equation since both
is in the nullspace of .
is chosen so that has a nullspace.
Of course, every matrix has a nullspace, but we want a nonzero eigenvector
(2) Checks on Eigenvalues
The sum of eigenvalues equals the sum of the diagonal entries:
Furthermore, the product of eigenvalues equals the determinant of
From example** of eigenvalues, the diagonal entries and the eigenvalues are the same only in triangular matrices. Normally, they are completely different.
- Everything is clear when
is a diagonal matrix:
The action of
- The eigenvalues of a projection matrix are 1 or 0.
A zero eigenvalue signals that
- **The eigenvalues are on the main diagonal when
This follows from property* of determinants.
The eigenvalues are
2. Diagonalization of a Matrix
(1) Eigenvectors Diagonalize a Matrix
matrix has linearly independent eigenvectors. If these vectors are the columns of a matrix , then is a diagonal matrix . The eigenvalues of are on the diagonal of :
the "eigenvector matrix" and the "eigenvalue matrix".
We split the matrix
- If a matrix has no repeated eigenvalues, then its eigenvectors are automatically independent.
Any matrix with distinct eigenvalues can be diagonalized.
- The diagonalizing matrix
is not unique.
- The order of the eigenvectors in
and the eigenvalues in is automatically the same.
- Not all matrices possess n linearly independent eigenvectors.
Not all matrices are diagonalizable.
depends on enough eigenvectors. (n independent eigenvectors)
depends on nonzero eigenvalues. (no zero eigenvalues)
- There is no connection between diagonalizability and invertibility.
- Diagonalization can fail only if there are repeated eigenvalues.
But it does not always fail.
3. Power and Products
- The eigenvalues of
has the same eigenvectors as , and eigenvalues .
is invertible, the eigenvalues of are .
Chapter 6 Positive Definite Matrices
1. Minima, Maxima and Saddle Points