Chapter 23
- Paragraph 1 – Definitions and examples
Let A be a square matrix.
An eigenvalue of A = a number r, which when subtracted from each of the diagonal entries of A,
converts A into a singular matrix
Note: a matrix is singular if and only if the determinant of the matrix is zero
r is an eigenvalue of 𝐴 if and only if 𝐴 – 𝑟𝐼 is a singular matrix:
det(𝐴 – 𝑟𝐼) = 0.
For a n x n matrix A, the left-hand side of the equation is an nth order polynomial in the variable r,
called the characteristic polynomial of A. An nth order polynomial has at most n roots and exactly n
roots if one counts roots with their multiplicity and complex roots.
So, an n x n matrix has at most n eigenvalues.
Theorem 23.1
The diagonal entries of a diagonal matrix D are the eigenvalues of D
Theorem 23.2
A square matrix A is singular if and only if 0 is an eigenvalue of A
A matrix M whose entries are nonnegative and whose columns (or rows) each add to 1 is called a
1 2
Markov matrix, e.j. (43 3
1). If we subtract a 1 from each diagonal entry of the Markov matrix, then
4 3
3 2
−4 3
each column of the transformed matrix adds up to 0, M − 1𝐼 = ( 3 2). If the columns of a
−3
4
square matrix add up to (0, …, 0) the rows are linearly dependent and the matrix must be singular.
So, r = 1 is an eigenvalue of every Markov matrix.
Recall: a square matrix B is non-singular if and only if the only solution of Bx = 0 is x = 0.
And B is singular if and only if the system Bx = 0 has a nonzero solution.
The fact that the square matrix 𝐴 – 𝑟𝐼 is a singular matrix when r is an eigenvalue of A means that
the system of equations (𝐴 – 𝑟𝐼)𝒗 = 𝟎 has a solution different from v = 0.
When r is an eigenvalue of A, a nonzero vector v such that (𝐴 – 𝑟𝐼)𝒗 = 𝟎 is called an eigenvector of
A corresponding to the eigenvalue r.
Theorem 23.3
Let A be an n x n matrix and let r be a scalar. Then, the following statements are equivalent:
1) Subtracting r from each diagonal entry of A transforms A into a singular matrix
2) 𝐴 – 𝑟𝐼 is a singular matrix
3) det(𝐴 – 𝑟𝐼) = 0
4) (𝐴 – 𝑟𝐼)𝒗 = 𝟎 for some nonzero vector v
5) 𝐴𝒗 = 𝑟𝒗 for some nonzero vector v
The set of all solutions to (𝐴 – 𝑟𝐼)𝒗 = 𝟎, including v = 0, is called the eigenspace of A with respect to
the corresponding eigenvalue.
Note: the eigenvalues of an upper- or lower-triangular matrix are precisely its diagonal entries
Note: for an invertible matrix A, r is an eigenvalue of A if and only if 1/r is an eigenvalue of 𝐴−1
Note: 𝐴−1 𝐴 = 𝐼
- Paragraph 1 – Definitions and examples
Let A be a square matrix.
An eigenvalue of A = a number r, which when subtracted from each of the diagonal entries of A,
converts A into a singular matrix
Note: a matrix is singular if and only if the determinant of the matrix is zero
r is an eigenvalue of 𝐴 if and only if 𝐴 – 𝑟𝐼 is a singular matrix:
det(𝐴 – 𝑟𝐼) = 0.
For a n x n matrix A, the left-hand side of the equation is an nth order polynomial in the variable r,
called the characteristic polynomial of A. An nth order polynomial has at most n roots and exactly n
roots if one counts roots with their multiplicity and complex roots.
So, an n x n matrix has at most n eigenvalues.
Theorem 23.1
The diagonal entries of a diagonal matrix D are the eigenvalues of D
Theorem 23.2
A square matrix A is singular if and only if 0 is an eigenvalue of A
A matrix M whose entries are nonnegative and whose columns (or rows) each add to 1 is called a
1 2
Markov matrix, e.j. (43 3
1). If we subtract a 1 from each diagonal entry of the Markov matrix, then
4 3
3 2
−4 3
each column of the transformed matrix adds up to 0, M − 1𝐼 = ( 3 2). If the columns of a
−3
4
square matrix add up to (0, …, 0) the rows are linearly dependent and the matrix must be singular.
So, r = 1 is an eigenvalue of every Markov matrix.
Recall: a square matrix B is non-singular if and only if the only solution of Bx = 0 is x = 0.
And B is singular if and only if the system Bx = 0 has a nonzero solution.
The fact that the square matrix 𝐴 – 𝑟𝐼 is a singular matrix when r is an eigenvalue of A means that
the system of equations (𝐴 – 𝑟𝐼)𝒗 = 𝟎 has a solution different from v = 0.
When r is an eigenvalue of A, a nonzero vector v such that (𝐴 – 𝑟𝐼)𝒗 = 𝟎 is called an eigenvector of
A corresponding to the eigenvalue r.
Theorem 23.3
Let A be an n x n matrix and let r be a scalar. Then, the following statements are equivalent:
1) Subtracting r from each diagonal entry of A transforms A into a singular matrix
2) 𝐴 – 𝑟𝐼 is a singular matrix
3) det(𝐴 – 𝑟𝐼) = 0
4) (𝐴 – 𝑟𝐼)𝒗 = 𝟎 for some nonzero vector v
5) 𝐴𝒗 = 𝑟𝒗 for some nonzero vector v
The set of all solutions to (𝐴 – 𝑟𝐼)𝒗 = 𝟎, including v = 0, is called the eigenspace of A with respect to
the corresponding eigenvalue.
Note: the eigenvalues of an upper- or lower-triangular matrix are precisely its diagonal entries
Note: for an invertible matrix A, r is an eigenvalue of A if and only if 1/r is an eigenvalue of 𝐴−1
Note: 𝐴−1 𝐴 = 𝐼
