Math 213 Chapter 4

If lambda is an eigenvalue of A, what is the determinant of (A - lambda I)?

0

Another way to state diagonalization

1 <= dim(lambda eigenspace) <= algebraic multiplicity of lambda for each eigenvalue lambda, and A will be diagonalizable if and only if dim( lambda eigenspace) = algebraic multiplicity of lambda for each lambda.

Diagonalization Steps

1. Find the eigenvalues of A. 2. Find linearly independent eigenvectors of A (if they exist). 3. Construct the matrix P whose columns are the above eigenvectors. 4. Construct the diagonal matrix D from the corresponding eigenvalues of D (in the same order as step 3). 5. Check that A = PD(P^-1).

Theorem: Let A be a square matrix with eigenvalue lambda and corresponding eigenvector x.

1. For any positive integer n, the value lambda^n is an eigenvalue of A^n with eigenvector x. 2. If A is invertible, then 1/(lambda) is an eigenvalue of A^-1 with eigenvector x.

When will a scalar lambda be an eigenvalue of matrix A?

1. If and only if it satisfies the characteristic equation of A, or 2. If and only if it is a root of the characteristic polynomial of A.

Facts about determinants when A is a square matrix

1. If one of the row (or columns) of A is zero, then detA = 0. 2. If A has two rows (or columns) that are the same, then detA = 0. 3. If a multiple of one row is added to another row to produce a matrix B, then detB = detA. 4. If two rows of A are interchanged to produce B then detB = -detA. 5. If one row of A is multiplied by k to produce B, then detB = k*detA.

Facts about geometric multiplicity given A is an n x n matrix.

1. The geometric multiplicity of lambda is less than or equal to its algebraic multiplicity. 2. A is diagonalizable if and only if the geometric multiplicities add up to n. 3. If A is diagonalizable then combining bases for each eigenspace gives a basis for all of R^n.

Equivalent facts about eigenvalues

1. lambda is an eigenvalue of A. 2. There is a nonzero vector x such that Ax = lambda x. 3. There is a nonzero vector x such that (A - lambda I) x = 0. 4. The matrix A - lambda I is NOT invertible. 5. det(A - lambda I) = 0.

If lambda = 0 is an eigenvalue of a square matrix A...

A is NOT invertible

Diagonalizable definition

A matrix is said to be diagonalizable if it is similar to some diagonal matrix D. In other words, the matrix A is diagonalizable if there is some invertible matrix P so that A = PD(P^-1) for some diagonal matrix D.

Eigenvalue definition

A scalar lambda is called an eigenvalue of A if there is a nontrivial solution x of Ax = lambda x; such an x is called an eigenvector corresponding to lambda

Eigenvector definition

An eigenvector of an n x n matrix A is a nonzero vector x such that Ax = lambda x for some scalar lambda.

The Diagonalization theorem

An n x n matrix A is only diagonalizable if it has n linearly independent eigenvectors. Moreover, the columns of P must consist of the eigenvectors of A. The diagonal entries of D will be the eigenvalues of A in the same order as the eigenvector columns of P.

Diagonalizable shortcut

An n x n matrix with n distinct eigenvalues is diagonalizable. WARNING: Just because a matrix doesn't have n distinct eigenvalues doesn't mean it's not diagonalizable.

Determinant definition

For a 1 x 1 matrix A = [a], the determinant of A, denoted by det(A), is defined to be det(A) = a. For n >= 2, the determinant of an n x n matrix is using expansion alternating positive and negative signs.

Characteristic Equation definition

For an n x n matrix A, the equation det(A - lambda I) = 0.

Power Rule for lambda and its eigenvector x.

If you raise lambda to the power of n, then A is also raised to the power of n and vice versa. This includes inversion (negative powers) if A is invertible.

If lambda is an eigenvalue of A, is the matrix A - lambda I invertible?

No

If two matrices have the same characteristic polynomial, does that imply that they are similar?

No

Is similarity the same as row equivalence?

No. In general, row operations may change eigenvalues.

Which matrices can have eigenvectors?

Only n x n matrices

Similar Matricies definition

Suppose A and B are n x n matrices, and that there exists an invertible matrix P such that B = (P^-1)AP. Then we say that A is similar to B

Cofactor definition

The cofactor is just a piece of the expanded determinant (i.e. a scalar multiplied by a portion of the original matrix)

What are the eigenvalues of a triangular matrix?

The entries on its main diagonal

Geometric Multiplicity definition

The geometric multiplicity of an eigenvalue is the dimension of its eigenspace.

Algebraic multiplicity definition

The multiplicity as a root of the characteristic polynomial of a particular eigenvalue

The determinant of a triangular matrix

The product of the entries on the main diagonal of the matrix.

Eigenspace definition

The set of all solutions to the equation (A - lambda I)x = 0 is called the eigenspace of A corresponding to lambda

If v1, ..., vr are eigenvectors corresponding to distinct eigenvalues lambda 1, ..., lambda r, then...

The set {v1,...,vr} is linearly independent.

Characteristic Polynomial definition

The term det(A - lambda I) which will be a polynomial in the variable lambda

If two matrices are similar then what does that mean?

The two matrices have the same characteristic polynomial, and hence the same eigenvalues (with the same multiplicities).

Are similar matrices commutative?

Yes. In other words, if A is similar to B, then B is similar to A, and we say that A and B are similar matrices.

If A and B are n x n matrices, then it's often NOT true that

det(A+B) != detA + detB

If A and B are n x n matrices, then it's true that

det(AB) = (detA)(detB).

If A is an invertible matrix, then

det(A^-1) = 1/(detA)

If A is an n x n matrix, then det(A^T) = ?

detA

A square matrix is invertible if and only if

detA != 0

If the columns of A span R^n

detA != 0

If the rows of A span R^n

detA != 0

Find the determinant when A is a square matrix and A has two rows/columns that are the same

detA = 0

Find the determinant when A is a square matrix and one of the rows/columns of A is zero

detA = 0

If the columns of A are linearly dependent, then

detA = 0

If the rows of A are linearly dependent, then

detA = 0

Find the determinant when A is a square matrix and two rows of A are interchanged to produce B

detB = -detA

Find the determinant when A is a square matrix and a multiple of one row is added to another row to produce a matrix B

detB = detA

Find the determinant when A is a square matrix and one row of A is multiplied by k to produce B

detB = k*detA