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Eigenvalues and eigenvectors
Chapters 7-8: Linear Algebra
Sections 7.5, 7.8 & 8.
Chapters 7-8: Linear Algebra
Linear systems of equationsInverse of a matrix Eigenvalues and eigenvectors
DeÖnitions Solutions
1. Linear systems of equations
A linear system of equations of the form
a 11 x 1 + a 12 x 2 + + a 1 n xn = b 1 a 21 x 1 + a 22 x 2 + + a 2 n xn = b 2 am 1 x 1 + am 2 x 2 + + amn xn = bm
can be written in matrix form as AX = B, where
A =
a 11 a 12 a 1 n a 21 a 22 a 2 n .. .
am 1 am 2 amn
, X =
x 1 x 2 .. . xn
, B =
b 1 b 2 .. . bm
Eigenvalues and eigenvectors
DeÖnitions Solutions
Solution(s) of a linear system of equations
(^1) Given a matrix A and a vector B, a solution of the system AX = B is a vector X which satisÖes the equation AX = B. (^2) If B is not in the column space of A, then the system AX = B has no solution. One says that the system is not consistent. In the statements below, we assume that the system AX = B is consistent. (^3) If the null space of A is non-trivial, then the system AX = B has more than one solution. (^4) The system AX = B has a unique solution provided dim(N (A)) = 0. (^5) Since, by the rank theorem, rank(A) + dim(N (A)) = n (recall that n is the number of columns of A), the system AX = B has a unique solution if and only if rank(A) = n.
Chapters 7-8: Linear Algebra
Linear systems of equationsInverse of a matrix Eigenvalues and eigenvectors
DeÖnitions Solutions
Row operations.
There are three types of row operations: (^1) Multiply a nonzero constant times an entire row. (ri! ari ) (^2) Exchange rows. (ri! rj and rj! ri ) (^3) Add a multiple of one row to another. (ri! arj + ri ) Row operations do not change the span of the row space. There are corresponding column operations, which do not change the column space.
Eigenvalues and eigenvectors
DeÖnitions Solutions
Row operations to compute the rank of a matrix.
Given a matrix A, row operations do not change the row space. Since the matrix
A =
A
can be made into the matrix
A^0 =
A
by doing row operations, the two matrices have the same row spaces. It is easy to see that the Örst two rows are linearly independent, so the rank is 2. Chapters 7-8: Linear Algebra
Linear systems of equationsInverse of a matrix Eigenvalues and eigenvectors
DeÖnitions Solutions
Consistency
The system AX = B is consistent, i.e., has a solution if (equivalently): (^1) Gaussian elimination on the augmented matrix (AjB) yields a matrix of the form: 0 BB BB BB BB @
a 1 j b 1 0 a 2 j b 2 0 0 0 a 3 j ... 0 0 0 0 ... j 0 0 0 0 0 0 ar j br 0 0 0 0 0 0 0 0 j 0 0 0 0 0 0 0 0 0 j 0
1 CC CC CC CC A
,
i.e., any rows reduced to all zeroes before the line are also zero after the line. (^2) The rank of (AjB) is equal to the rank of A.
Eigenvalues and eigenvectors
DeÖnitions Solutions
Inconsistency
The system AX = B is inconsistent, i.e., has NO SOLUTION if (equivalently): (^1) Gaussian elimination on the augmented matrix (AjB)yields a matrix of the form: 0 BB BB BB BB @
a 1 j b 1 0 a 2 j b 2 0 0 0 a 3 j ... 0 0 0 0 ... j 0 0 0 0 0 0 ar j br 0 0 0 0 0 0 0 0 j br + 1 0 0 0 0 0 0 0 0 j 0
1 CC CC CC CC A
,
where br + 1 6 = 0 , i.e., there is a row of zeroes before the line with a nonzero element after the line. (^2) The rank of (AjB) is greater than the rank of A. (^3) The vector B is not in the column space of A. Chapters 7-8: Linear Algebra
Linear systems of equationsInverse of a matrix Eigenvalues and eigenvectors
DeÖnitions Solutions
Unique solutions
The system AX = B has one unique solution if (equivalently): (^1) Gaussian elimination on the augmented matrix (AjB) yields a matrix of the form: 0 BB BB BB B B@
a 1 j b 1 0 a 2 j b 2 0 0 a 3 j ... 0 0 0 ... j 0 0 0 0 an j bn 0 0 0 0 0 j 0 0 0 0 0 0 j 0
1 CC CC CC C CA
,
i.e., there are all nonzero numbers on the ìdiagonal.î (^2) The rank of A is equal n (which is equal to the rank of (AjB)), which is the maximum rank (so it is essential that n m). This means that dim(N (A)) = 0 , i.e., the nullspace is trivial. (^3) The columns of A form a basis for the column space.
Linear systems of equationsInverse of a matrix Eigenvalues and eigenvectors
DeÖnitions Determinant of a matrix Properties of the inverseLinear systems of n equations with n unknowns
2. Inverse of a matrix
If A is a square n n matrix, its inverse, if it exists, is the matrix, denoted by A ^1 , such that A A ^1 = A ^1 A = In ,
where In is the n n identity matrix. A square matrix A is said to be singular if its inverse does not exist. Similarly, we say that A is non-singular or invertible if A has an inverse. The inverse of a square matrix A = [aij ] is given by
A ^1 =
det(A)
[Cij ]T^ ,
where det(A) is the determinant of A and Cij is the matrix of cofactors of A. Chapters 7-8: Linear Algebra
Linear systems of equationsInverse of a matrix Eigenvalues and eigenvectors
DeÖnitions Determinant of a matrix Properties of the inverseLinear systems of n equations with n unknowns
Determinant of a matrix
The determinant of a square n n matrix A = [aij ] is the scalar det(A) =
n
i = 1
aij Cij =
n
j = 1
aij Cij
where the cofactor Cij is given by
Cij = ( 1 )i^ +j^ Mij ,
and the minor Mij is the determinant of the matrix obtained from A by ìdeletingî the i-th row and j-th column of A.
Example: Calculate the determinant of A =
Linear systems of equationsInverse of a matrix Eigenvalues and eigenvectors
DeÖnitions Determinant of a matrix Properties of the inverseLinear systems of n equations with n unknowns
Properties of determinants
If a determinant has a row or a column entirely made of zeros, then the determinant is equal to zero.
The value of a determinant does not change if one replaces one row (resp. column) by itself plus a linear combination of other rows (resp. columns).
If one interchanges 2 columns in a determinant, then the value of the determinant is multiplied by 1.
If one multiplies a row (or a column) by a constant C , then the determinant is multiplied by C.
If A is a square matrix, then A and AT^ have the same determinant.
Chapters 7-8: Linear Algebra
Linear systems of equationsInverse of a matrix Eigenvalues and eigenvectors
DeÖnitions Determinant of a matrix Properties of the inverseLinear systems of n equations with n unknowns
Properties of the inverse
Since the inverse of a square matrix A is given by
A ^1 =
det(A)
[Cij ]T^ ,
we see that A is invertible if and only if det(A) 6 = 0.
If A is an invertible 2 2 matrix,
a 11 a 12 a 21 a 22
, then
A ^1 =
det(A)
a 22 a 12 a 21 a 11
and det(A) = a 11 a 22 a 21 a 12. If A and B are invertible, then (AB) ^1 = B ^1 A ^1 and
A ^1
= A.
Linear systems of equationsInverse of a matrix Eigenvalues and eigenvectors
DeÖnitions Determinant of a matrix Properties of the inverseLinear systems of n equations with n unknowns
Linear systems of equations - summary (continued)
Consider the linear system AX = B where A is an m n matrix.
If m = n and the system is consistent, then Either det(A) 6 = 0, in which case rank(A) = n, dim(N (A)) = 0, and the system has a unique solution; Or det(A) = 0, in which case dim(N (A)) > 0, rank(A) < n, and the system has an inÖnite number of solutions.
Note that when m = n, having det(A) = 0 means that the columns of A are linearly dependent. It also means that N (A) is non-trivial and that rank(A) < n.
Chapters 7-8: Linear Algebra
Linear systems of equationsInverse of a matrix Eigenvalues and eigenvectors
Eigenvalues Eigenvectors
3. Eigenvalues and eigenvectors
Let A be a square n n matrix. We say that X is an eigenvector of A with eigenvalue λ if
X 6 = 0 and AX = λ X.
The above equation can be re-written as
(A λ In )X = 0.
Since X 6 = 0, this implies that A λ In is not invertible, i.e. that det(A λ In ) = 0.
The eigenvalues of A are therefore found by solving the characteristic equation det(A λ In ) = 0.
Eigenvalues and eigenvectors
Eigenvalues Eigenvectors
Eigenvalues
The characteristic polynomial det(A λ In ) is a polynomial of degree n in λ. It has n complex roots, which are not necessarily distinct from one another.
If λ is a root of order k of the characteristic polynomial det(A λ In ), we say that λ is an eigenvalue of A of algebraic multiplicity k.
If A has real entries, then its characteristic polynomial has real coe¢ cients. As a consequence, if λ is an eigenvalue of A, so is λ ¯.
It A is a 2 2 matrix, then its characteristic polynomial is of the form λ^2 λ Tr(A) + det(A), where the trace of A, Tr(A), is the sum of the diagonal entries of A. Chapters 7-8: Linear Algebra
Linear systems of equationsInverse of a matrix Eigenvalues and eigenvectors
Eigenvalues Eigenvectors
Eigenvalues (continued)
Examples: Find the eigenvalues of the following matrices.
A =
1 0 0 5
.
B =
1 9 0 5
.
C =
13 36 6 17
.
D =
2 4
4 1 1 1 4 1 1 1 2
3 (^5).
