Let A be a 2 ×2 matrix, and call a line through the origin...

Let $A$ be a $2 \times 2$ matrix, and call a line through the origin of $R^{2}$ invariant under $A$ if $A \mathbf{x}$ lies on the line when x does. Find equations for all lines in $R^{2}$, if any, that are invariant under the given matrix. (a) $A=\left[\begin{array}{rr}4 & -1 \\ 2 & 1\end{array}\right]$ (b) $A=\left[\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right]$

Let $A$ be a $2 \times 2$ matrix, and call a line through the origin of $R^{2}$ invariant under $A$ if $A \mathbf{x}$ lies on the line when x does. Find equations for all lines in $R^{2}$, if any, that are invariant under the given matrix.
(a) $A=\left[\begin{array}{rr}4 & -1 \\ 2 & 1\end{array}\right]$
(b) $A=\left[\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right]$

Key Concepts

Invariant Subspaces

An invariant subspace under a linear transformation is a subspace that, when any vector within it is transformed, remains within the same subspace. In R², a line through the origin is a one?dimensional invariant subspace if every vector on that line is mapped to another vector lying on the same line. This concept is central to understanding how linear maps behave on subsets of the vector space, and in many problems, finding invariant subspaces simplifies the analysis of the transformation.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental elements in linear algebra that help characterize invariant lines. An eigenvector of a matrix is a nonzero vector that, when multiplied by the matrix, results in a scalar multiple of itself, where the scalar is the eigenvalue. The line spanned by an eigenvector is invariant under the matrix because every vector on that line is just a scalar multiple of the eigenvector and thus remains on the same line when the matrix is applied.

Rotation Matrices

Rotation matrices are specific types of matrices that represent rotations in the plane. A rotation matrix typically does not leave any nontrivial one?dimensional subspace invariant (except in the trivial cases of a 0° or 180° rotation) since it rotates every nonzero vector to a different direction. In problems involving rotation matrices, the lack of real eigenvalues (or the presence of complex eigenvalues) indicates that no real line through the origin remains invariant under the transformation.

Transcript

0:00 Hello there.

00:01 Okay, so for this exercise, we need to find all the lines that are going to be invariant to certain transformation into 2.

00:08 And these lines should pass through the origin.

00:11 Basically, what we need to find is all the lines that are going to be...

00:17 We need to find the solutions for the following system.

00:27 A .x equals to some linear combination of x.

00:32 Okay, basically, here we have some vector x, and after we apply the matrix a, we maintain invariant, right? we just extend or make the line smaller or we change the direction, basically.

00:47 That's what is going to do this lambda here.

00:51 So the idea here is that the lines that are going to be invariant under a are the eigenvectors.

01:06 The eigenvectors have the direction that the lines should fall from the origin to satisfy this condition.

01:15 Because if you apply to any point in that line, the matrix a, you will end in the same line.

01:23 Either you in a farther point from the origin or closer on the other direction and so on.

01:33 So that's the idea here.

01:35 So the statement, just say, to find the eigenvectors of the.

01:38 Of this matrix.

01:41 But to find the eigenvectors we need to find first the eigenvalues.

01:46 So the first thing is to find the eigenvalues and we know the procedure.

01:53 We need to obtain the characteristic equation that is obtained by the determinant of lambda times the entity in this case is the identity 2 by 2 minus the matrix equals to 0.

02:05 Okay so this determinant will be lambda minus 4 1 2 and lambda minus 1 here minus 2 this determinant give us the following quadratic polynomial that is lambda square minus 5 lambda plus 6 equals to 0 and we can write this as lambda minus 3 times lambda minus 2 equals to 0...

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