Haowen! Geometric and physical meanings of feature values and feature vectors]

We know that matrix multiplication corresponds to a transformation, which converts any vector into a new vector with different directions or lengths. During this transformation, the original vector mainly changes in rotation and scaling. If a matrix only performs scaling transformation on a vector or some vectors and does not rotate these vectors, these vectors are called feature vectors of this matrix, and the scaling ratio is the feature value.

In fact, the above section not only describes the matrix transformation feature value and the geometric meaning of the feature vector (graphical transformation) but also its physical meaning. Physical means the motion picture: the feature vector is scaled under the action of a matrix, and the scaling amplitude is determined by the feature value. If the feature value is greater than 1, all feature vectors that belong to this feature value have a violent length. If the feature value is greater than 0 and less than 1, the feature vectors have been quickly scaled down. If the feature value is smaller than 0, the feature vectors have shrunk over the border, the opposite direction goes.

Note: in textbooks, feature vectors are vectors that do not change the direction under matrix transformation. In fact, when the feature value is smaller than zero, the matrix changes the feature vectors completely in the inverse direction, of course, the feature vector is also the feature vector. I agree with the saying that feature vectors do not change the direction: feature vectors never change the direction, but only the feature values (the feature value of the direction reversal is a negative value ). It is similar to saying that the outdoor "temperature" in Shenzhen is 10 ℃ in winter, and that in Harbin is-30 ℃ (temperature rather than temperature ); it is also similar to saying that an unmanned aircraft is flying at an altitude of 100 meters, while a nuclear submarine is swimming at an altitude of 50 meters.

Note the two highlights of feature values and feature vectors. One of the two highlights is the meaning of linear invariant, and the other is the meaning of the vibration spectrum.

Feature vectors are linear constants.

 

 

One of the highlights of the so-called feature vector concept is the invariant, which is called linear invariant. As we often say, linear transformation and linear transformation do not mean turning a line (vector) into another line (vector ), most of the changes in the line are changes in the direction and length. However, a vector named "feature vector" is special, and only the length is changed in the direction of the matrix. The constant direction is called linear invariant.

If some readers insist that the feature vector in the negative direction is the idea of changing the vector direction, you may consider linear invariant as follows: the invariant feature vectors are the vectors that are collocated with themselves, and their straight lines remain unchanged under linear transformation; the feature vectors and Their transformed vectors are in the same straight line. The transformed vectors can be extended or shortened, or reverse, it even becomes a zero vector (the feature value is zero), for example.

 

The feature value is the vibration spectrum.

In addition to linear constants, another highlight is about vibration. In the Song Dynasty, China passed the opportunity to discover the matrix feature value theory. After Qin shaoyou threw a small stone into the pond, he just got a pick-up poem "throwing stones to the bottom of boiling water", and then the monkey hurried to the Cave Room, we did not think of the scientific principle of the matrix feature values and feature vectors hidden in water waves. Generally speaking, any of the water beads near the water surface vibrates up and down in the original place (in fact, it is doing an approximate circular motion) and does not move toward the outer ring with the waves, at the same time, the amplitude of the above and below vibration drops gradually until it becomes calm. In a matrix determined by a rock with a specific quality and shape being invested in an area and depth-specific pool at a certain angle and speed, the feature value plays a decisive role in the gradient process of water beads in ripple. It determines the frequency and the decline rate of the vibration of water beads.

In understanding the feature values and feature vectors of vibration, the concept of complex vectors and complex matrices must be added, because in actual application, real vectors and real matrices cannot do much. Mechanical Vibration and electrical vibration have a spectrum, and a frequency of vibration has a certain amplitude. Then the Matrix also has a matrix spectrum. The matrix spectrum is the concept of the matrix feature value and is inherent in the matrix, all the feature values form a spectrum of the matrix. Each feature value is a "resonant frequency" of the matrix ".

In his classic textbook linear algebra and its application, the American mathematician G. Strang introduced the physical significance of the feature value as a frequency. He said:

The simplest example (I never believe in its authenticity, although it is said that a bridge was broken down in 1831) is an example of a soldier passing through a bridge. Traditionally, they have to stop moving forward and walk through. The reason is that they may walk along at a frequency equal to one of the characteristic values of the bridge, which will cause resonance. Just like a child's swing, once you notice the frequency of a swing that matches the frequency, you make the frequency swing higher. An engineer is always trying to keep his bridge or rocket's natural frequency away from the frequency of the wind or the frequency of the liquid fuel; and in another extreme situation, A securities broker is dedicated to reaching the natural frequency line of the market. Feature value is the most important feature of almost any dynamic system.

In fact, the reason why this matrix can form a "frequency spectrum" is that the matrix has a constant transformation effect on the vector in the direction indicated by the feature vector: enhancement (or weakening) feature vectors. Further, if the matrix is continuously stacked to act on the vector, the feature vector will be highlighted.

For example, a physical system can be described by a matrix, so the physical characteristics of this system can be determined by the feature values of this matrix, a variety of different signals (vectors) after entering this system, the signal (vector) output by the system will be subject to various chaotic changes such as phase lag, amplification, and reduction. However, only feature signals (feature vectors) are steadily amplified (or reduced. If the system's output port is connected to the input port, only the feature signal (feature vector) is amplified (or reduced) for the second time, other signals, such as lagging signals, may lag ahead and be scaled down at the same time, while amplification may be scaled down or delayed at the same time, the scale-out may be further zoomed in or delayed. After N cycles, it is clear that a large number of chaotic vectors cannot form a climate. Only the feature vectors can think about each other and try their best to make a difference, you can either fail to gain benefits. Therefore, we can observe the output in the time domain and obtain one or more super obvious feature signals (feature vectors ).

The friends who have made the circuit have seen the influence: Cut! What do you mean about the oscillator? The oscillator signal (voltage and current) forms a feature vector. The feature value is 1, and the frequency of the oscillator signal is...

Yes, yes, that is, the principle of the oscillator. In fact, the oscillator principle can be explained by the power of the matrix. This editor is not easy to use. Matrix Analysis and details are ignored here.