Show that A is non defective and use Theorem 7.4 .3 to find e^A t. A=[ 1 2 0 3

step 1 we are given a matrix a. We are asked to show that this matrix is non -defective, and then to fi

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

Key Concepts

Non Defective Matrix

A matrix is called non defective if it has a complete set of linearly independent eigenvectors. This means its algebraic multiplicity equals its geometric multiplicity for each eigenvalue, allowing the matrix to be diagonalized. In turn, diagonalization simplifies computations such as evaluating matrix functions like the exponential.

Eigenvalues and Eigenvectors

Eigenvalues are scalars associated with a matrix that, when subtracted from the diagonal entries and solved in the determinant equation, yield nontrivial solutions for the corresponding eigenvectors. Eigenvectors are nonzero vectors that change only by a scalar factor when the matrix is applied. Finding them is central to understanding the behavior of the matrix and serves as the basis for diagonalization.

Diagonalization

Diagonalization is the process of finding an invertible matrix composed of the eigenvectors of the given matrix, which transforms the matrix into a diagonal form containing its eigenvalues. This process is crucial for computing functions of matrices, such as the exponential function, since operations on diagonal matrices are much simpler.

Matrix Exponential

The matrix exponential is a function that generalizes the scalar exponential function to square matrices. It is defined by the power series expansion and plays a key role in solving systems of linear differential equations. When a matrix is diagonalizable, the exponential can be computed by exponentiating each eigenvalue, making the process straightforward.

Theorem on Matrix Exponential (Theorem 7.4.3)

This theorem states that if a matrix is diagonalizable (non defective), its exponential can be computed by first diagonalizing the matrix, applying the exponential function to the diagonal eigenvalues, and then transforming back using the inverse of the diagonalizing matrix. This theorem provides a systematic approach to calculate e^(At), leveraging the ease of exponentiating a diagonal matrix.

Transcript

00:01 We are given a matrix a.

00:02 We are asked to show that this matrix is non -defective, and then to find e to the a -t.

00:13 So our matrix a is the 2x2 matrix with entries 1 ,203, and therefore the characteristic polynomial of a, which i'll call p of lambda, is equal to 1 minus lambda, 3 minus lambda minus 2 times 0, just simply 0.

00:50 So when our characteristic polynomial is equal to 0, we have our characteristic equation.

00:57 And we see that the solutions to this equation are the eigenvalues lambda 1 equals 1, and lambda 2 equals 3.

01:12 If v1 is an eigenvector at a with eigenvalue the lambda 1.

01:18 So first of all, because we have eigenvalues, it follows that a is non -defective.

01:39 And if we have an identector v1 with eigenvalue lambda 1, then it follows that a minus lambda 1 i v1 is equal to the 0 vector.

01:58 So this implies that a minus identity matrix, which is simply 0 to 0 ,0, 2, times our eigenvector with the.

02:23 V1 and 1 2 is equal to the 0 matrix.

02:31 And this then implies that v1 is equal to 0.

02:41 This is what both equations tell us.

02:44 And therefore we can have that v11 can be any real number.

02:49 Let's just pick v1 to v1.

02:53 Then we obtain that our function, that our eigenvector v1 is the vector 1 -0.

03:05 If v2 is an eigenvector for eigenvalue lambda 2, then we have that a minus lambda 2 ib2 is equal to the 0 vector.

03:20 And since lambda 2 is equal to 3 follows that this matrix is going to be 1 minus 3 or negative 2, 0, and 3 minus 3, which is another 0, times our vector v2, which is components v21, b2, is equal to the 0, is equal to the 0 matrix.

04:01 The zero vector.

04:07 So we get that negative 2 v21 plus 2 v2 is equal to 0.

04:20 Therefore we have that v22 is equal to v21...

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