Eigenvalues and Eigenvectors, Imaginary and Real

Physics Videos by Eugene Khutoryansky

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3D visualization of eigenvectors and eigenvalues. Includes imaginary and real components. My Patreon page is at https://www.patreon.com/EugeneK

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Eigenvalues and Eigenvectors play a critical role

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in determining observable measurements of quantum systems,

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in evaluating the stability of mechanical structures,

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in analyzing the feedback loops of electric circuits,

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and in many other areas.

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To fully understand physics and engineering,

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it is necessary to understand Eigenvalues and Eigenvectors.

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Suppose we have an object.

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Here we are viewing the object from different perspectives.

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Now, suppose we apply what we call a linear transformation.

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As we again view the object from different perspectives,

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we see that the object is now distorted.

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Let us consider the points on this object before the transformation.

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Each of these points can be represented by an arrow.

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When the transformation is applied, the length and direction of each arrow can change.

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There are some arrows that

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point along the same line both before and after the transformation.

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For example, these three arrows will simply

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have their lengths multiplied by a constant.

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The length of the orange arrow is multiplied by negative one.

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The length of the blue arrow is multiplied by two.

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And the length of the green arrow is multiplied by one half.

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We say that negative one, two, and one half

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are the “eigenvalues” of this transformation.

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The arrows that point along the same line both before and after the transformation

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are what we call the “eigenvectors.”

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The amounts by which the lengths of each of these arrows is multiplied

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are what we call the eigenvalues.

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The eigenvalues and eigenvectors

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dictate the nature of the transformation for the entire object.

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Consider, for example, the point symbolized by this white arrow.

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The white arrow is not aligned with the directions of any of the eigenvectors,

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which are symbolized by the orange, green, and blue lines.

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But, the white arrow can be thought of as a combination of arrows

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that are parallel to the directions of these three lines.

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When the transformation is applied, the lengths of each of these arrows

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is multiplied by the eigenvalue associated with that direction.

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That is, the orange, green, and blue arrows continue pointing in the same direction,

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but are multiplied by “-1”, “1/2”, and “2.”

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In this way, the transformation of any point on this object

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can be determined by using the eigenvalues and eigenvectors.

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The eigenvectors do not necessarily have to be 90 degrees to one another.

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For example, let us consider this new object, and this new transformation.

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The red and green arrows are eigenvectors of this transformation.

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The green arrow is multiplied by one, and the red arrow is multiplied by two.

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Now consider the transformation of this white arrow.

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The white arrow can be thought of as the

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combination of arrows parallel to the green and red lines.

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During the transformation, this green arrow is multiplied by one,

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and this red arrow is multiplied by two.

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The eigenvalues and eigenvectors do not necessarily have to consist of real numbers.

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The eigenvalues of this transformation are the imaginary numbers “i” and negative “i.”

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And the eigenvectors are the following.

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Let us say that the red arrow is signified by a “1” followed by a “0.”

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And let’s say that the yellow arrow is signified by a “zero” followed by a “one.”

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Before the transformation,

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the blue arrow is in the same direction as the red arrow, but double its length.

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Therefore, using the notation described, the blue arrow before the transformation

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would be signified by a “2” followed by a “0.”

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After the transformation, the blue arrow is pointed in the

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opposite direction of the yellow arrow, and double its length.

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Therefore, the blue arrow after the transformation

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would be signified by a “0” followed by a “-2.”

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Let us consider the blue arrow before the transformation.

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The blue arrow can be thought of as the sum of the two eigenvectors.

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Suppose we multiply each eigenvector by its corresponding eigenvalue.

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The result is the new direction of our blue arrow after the transformation.

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Therefore, everything discussed here is still true

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even for imaginary values for eigenvalues and eigenvectors.

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Much more information about linear transformations is available in the video

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“Linear Algebra – Matrix Transformations.”

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