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

step 1 We are given a matrix. We are asked to show the matrix is non -defective and to find E to the A

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

Key Concepts

Application of Theoretical Results (e.g., Theorem 7.4.3)

The use of specific theorems, such as Theorem 7.4.3, provides a systematic method to calculate the exponential of matrices, especially for 2x2 cases with complex eigenvalues. These theorems typically offer explicit formulas that involve the matrix's eigenvalues and the transformation matrix from its diagonalization. They are of particular importance in ensuring that the solution is valid and well-structured for matrices that are non-defective.

Non-defective Matrices

A non-defective matrix is one that has a complete set of linearly independent eigenvectors, which ensures that it is diagonalizable. This property is crucial because diagonalizable matrices allow the matrix exponential to be computed easily by transforming the matrix into a diagonal form. When a matrix is non-defective, powerful techniques such as those outlined in relevant theorems (e.g., Theorem 7.4.3) can be applied to obtain closed-form solutions.

Diagonalization

Diagonalization is the process of finding a similarity transformation that converts a matrix into a diagonal matrix. This concept is particularly useful when computing the exponential of a matrix since the exponential of a diagonal matrix involves exponentiating each diagonal entry individually. The process hinges on the matrix being non-defective, meaning it has enough independent eigenvectors to allow such a transformation.

Matrix Exponential

The matrix exponential, e^(At), is a fundamental concept in linear systems and differential equations. It generalizes the scalar exponential function to matrices and provides the solution to systems of linear differential equations. The exponential of a matrix is defined by the power series expansion and has numerous applications in solving initial value problems.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are critical in understanding the behavior of linear transformations represented by matrices. They are used to decompose a matrix into simpler components that reveal much about its structure, such as stability and oscillatory behavior in dynamical systems. In the context of the matrix exponential, knowing the eigenvalues is essential for applying formulas or theorems that simplify the computation.

Transcript

00:01 We are given a matrix.

00:03 We are asked to show the matrix is non -defective and to find e to the a -t.

00:12 Our matrix a is negative 1, 3, negative 3, negative 1.

00:20 The characteristic polynomial a is negative 1 minus lambda squared, minus 3 times negative 3, or plus 9.

00:35 And so the characteristic equation of a is this equal to zero.

00:44 So we get negative 1 minus lambda is equal to plus or minus 3i, or that lambda is equal to negative 1 plus or minus 3i.

01:09 So we have the eigenvalues for a of lambda 1 equals negative 1 minus 3i, and so lambda 2 equals negative 1 plus 3i.

01:29 If v1 is an eigenvector associated with the eigenvalue lambda 1, then we have that a minus lambda 1, or plus 1 plus 3i, i times v1 is equal to the 0 vector.

01:53 Therefore, we have that negative 1 plus 1 plus 3i, or 3i, 3, negative 3, and negative 1, again, plus 1 plus 3i, which is 3i, times b11, b12, is equal to 0.

02:23 Of 0.

02:30 Therefore, we have the 3i v11 plus 3 v12 is equal to 0.

02:45 So v12 is equal to negative i.

02:54 Therefore, if v11 is equal 1, then v12 is going to be negative i.

03:00 So we get that our eigenvector v1 is 1 negative i.

03:10 If v2 is an eigenvector, if v2 is an eigenvector associated with eigenvalue lambda 2, and we have that a minus lambda 2 i, which is minus negative 1 plus 3i, i v2 is equal to the 0 vector.

03:40 Therefore, negative 1 minus negative 1 plus 3i, which is negative 3i, 3, i 3, negative 3, negative 3 i v2 1 b22 is equal to 0 0 therefore we get the negative 3 i v2 1 plus 3 v2 2 2 is equal to 0 so the v22 2 2 is equal to i v2 and therefore v2 1 is equal to 1 v2 2 2 is equal to i so we get that an eigenvector v2 is 1i.

04:47 Now since we have these eigenvalues, we know that a is non -defective, and therefore we can find two matrices s and d, such that a is equal to s -d inverse, where s is the matrix whose column vectors are the eigenvectors of a...

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