(a) Write out the characteristic equation for the matrix ( ...

(a) Write out the characteristic equation for the matrix $\left(\begin{array}{ccc}0 & 1 & 0 \\ 0 & 0 & 1 \\ \alpha & \beta & \gamma\end{array}\right)$. (b) Show that, given any 3 numbers $a, b$, and $c$, there is a $3 \times 3$ matrix with characteristic equation $-\lambda^3+a \lambda^2+b \lambda+c=0$.

(a) Write out the characteristic equation for the matrix $\left(\begin{array}{ccc}0 & 1 & 0 \\ 0 & 0 & 1 \\ \alpha & \beta & \gamma\end{array}\right)$.
(b) Show that, given any 3 numbers $a, b$, and $c$, there is a $3 \times 3$ matrix with characteristic equation $-\lambda^3+a \lambda^2+b \lambda+c=0$.

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Key Concepts

Characteristic Polynomial

The characteristic polynomial of a matrix is obtained by computing the determinant of (?I ? A) and setting it equal to zero. It encapsulates information about the eigenvalues of the matrix, as they are the roots of this polynomial, and it plays a central role in understanding a matrix's algebraic properties, such as stability and diagonalizability.

Companion Matrix

A companion matrix is a specially structured square matrix whose characteristic polynomial is exactly a given monic polynomial. It is used to demonstrate that for any polynomial (with appropriate degree and coefficients) one can construct a matrix that has that polynomial as its characteristic equation, thereby linking polynomial theory with matrix theory.

Transcript

0:00 Hello there.

00:02 So for this exercise we got two polynomials, two characteristic polynomials.

00:08 This is the first one.

00:09 And based on the characteristic polynomial, we need to first determine the dimension or the size, properly said, of the associated matrix a.

00:22 And we need to determine the dimension of each of the eigenspaces.

00:31 For each eigenvalue.

00:34 We don't need to determine exactly the value of the dimension because we don't have enough information here, but at least what are the possible dimensions for the eigenspaces.

00:47 So let's start with the dimension, with the size of the matrix a, and that's determined by the degree of the point of note.

00:54 In this case, just for some notation, a subscript lambda i corresponds to the eigenspace of associated to that eigenvalue and this triangle here lambda, of course is the notation for the characteristic polynomial of the matrix 8.

01:14 So the degree of this polynomial in this case is equals to 1 -1 -1, so we need to sum this and this gives us 3.

01:26 So that's the degree of this polynomial and that implies that the size of a a is a 3 by 3 matrix.

01:39 Then we need to get the dimension.

01:42 So this is done.

01:44 Then we need to get the dimension of the eigenspaces.

01:51 And for that, in this case we got three eigenvalues...

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