Show that A is non defective and use Theorem 7.4 .3 to find e^A t. A=[ 3 1 1 3

Show that A is non defective and use Theorem 7.4 .3 to find...

Key Concepts

Theorem on Matrix Exponential (Theorem 7.4.3)

Theorem 7.4.3 provides a method for computing the exponential of a matrix, particularly when the matrix is non defective. The theorem leverages the diagonalizability of the matrix, allowing one to compute e^(At) easily by diagonalizing the matrix, computing the exponential of the diagonal matrix, and then transforming back via the similarity transformation. This approach substantially reduces the complexity involved in evaluating matrix exponentials.

Non defective matrix

A non defective matrix is one that has a complete set of linearly independent eigenvectors. This ensures that the matrix is diagonalizable, meaning it can be represented as PDP?¹, where D is a diagonal matrix of eigenvalues and P contains the corresponding eigenvectors. Diagonalizability is a crucial property for simplifying many matrix computations, including the matrix exponential.

Eigenvalues and Eigenvectors

Eigenvalues are scalars obtained from the equation Av = ?v, where A is a square matrix and v is a nonzero vector called the eigenvector. They reveal important properties of A, such as its scaling actions along particular directions. Finding eigenvalues and eigenvectors is essential for diagonalization and for understanding the dynamics described by the matrix.

Diagonalization

Diagonalization is the process of converting a matrix into a diagonal form via a similarity transformation. A matrix A can be written as A = PDP?¹ when it is diagonalizable, with D being a diagonal matrix whose entries are the eigenvalues of A. This transformation simplifies computations like raising the matrix to a power or taking its exponential, as operations on diagonal matrices are considerably easier.

Matrix Exponential

The matrix exponential e^(At) is a function defined by an infinite series analogous to the scalar exponential function. It is widely used for solving systems of linear differential equations. When A is diagonalizable, the computation of e^(At) simplifies to e^(At) = P e^(Dt) P?¹, where e^(Dt) is computed by exponentiating each diagonal entry of D, thus reducing the problem to handling scalar exponentials.

Transcript

00:01 We were given a matrix, and we are asked to show that this matrix is non -defective and to evaluate e to the a -t.

00:11 Our matrix a is 3113.

00:18 The characteristic polynomial of a is 3 minus lambda squared minus 1, and the characteristic equation is when this is equal to 0, so that we get 3 minus lambda square is equal to 1, more that 3 minus lambda square is equal to 1, more that 3.

00:40 Minus lambda equal to plus or minus one.

00:45 So that lambda is equal to the opposite of plus or minus one minus three, which is three minus plus plus one.

00:57 So that we get that our eigen values are lambda 1 equals 2 and lambda 2 equals 4.

01:15 Therefore, a is non -defective.

01:19 Suppose that v1 is the eigenvector associated with eigenvalue lambda 1, then we have that a minus 2i, v1 is equal to the zero vector.

01:51 Therefore we have that 3 minus 2, or 1, 1, 3 minus 2 again is 1 times our eigenvector, b11 v1 is equal to the zero vector zero zero and therefore we have that v11 plus b12 equals 0 so that v12 is the opposite of v11 if we take v1111 to v1 then b12 is equal to negative 1 so that our eigenvector v1 will be 1 negative 1 if v2 is the eigenvector associated with the eigenvalue lambda 2, then we have the a minus land of 2, which is 4, i times v2, equals the zero vector.

03:06 So that we get 3 minus 4, which is negative 1, 1, 3 minus 4 again, times v21, v222 is equal to the zero vector 0 -0.

03:29 So we get that negative v21 plus v22 equals 0, so that v22 is equal to v21...

Please give Ace some feedback