我一直以来最喜欢的MATLAB示例之一是eigshow。

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eigshow

Theeigshow很长一段时间以来一直是MATLAB的一部分，一直是我的最爱之一。

When you starteigshow, a 2-by-2 matrix is shown in the title bar. There is also a green vectorx和蓝色矢量Ax, the image ofxunder the mapping induced byA。上面的动画gif中显示的默认值是

格式rat， 格式袖珍的A = [1 3; 4 2]/4 x = [1; 0] Ax = A*x

A = 1/4 3/4 1 1/2 x = 1 0 Ax = 1/4 1

椭圆

现在移动xwith the mouse.xstays on the unit circle.Axgoes whereverAtakesx, generating the blue curve, an ellipse.

Line 'em up

Runeigshowyourself. Don't just watch the animation. The instructions at the bottom of the screen say "Make A*x parallel to x". When you do this -- when the blue and green vectors are pointing in the same direction -- you have found an eigenvector. The green vector is the eigenvector and the length of the overlapping blue vector is the eigenvalue.

Eigenvectors

One eigenvector of this particular matrix is in the first or third quadrant, making an angle of roughly 60 degrees with the horizontal axis. The blue vector is longer than the green. The eigenvalue is actually 5/4.

当蓝色矢量指向与绿色相反的方向时，获得另一个特征向量，因为特征值为负。特征向量位于第二或第四象限，角度约为45度。特征值为-1/2。

请注意，特征向量不是椭圆的主要和次要轴。该矩阵并不完全对称。如果是对称的，则特征向量将是轴。

A Second Matrix

The title bar ofeigshowis a pull-down menu that offers a dozen different matrices. Our second example swaps the top two elements of the first example.

a = [3 1;4 2]/4

A = 3/4 1/4 1 1/2

Now the aspect ratio of the ellipse is quite a bit larger. Both eigenvalues are positive. If you watch carefully, you can see thatAxgoes around in the same direction asx。Our first matrix has one negative eigenvalue and sox和Axtravel in opposite directions.

第三个矩阵

Now choose the matrix that swaps the bottom two elements and flips one sign.

a = [3 1;-2 4]/4

A = 3/4 1/4 -1/2 1

试图制作斧头parallel tox令人沮丧。你做不到。更确切地说，您必须从页面或屏幕上删除。该矩阵没有任何真实的特征值和特征向量。它的特征值和特征向量很复杂。

稍后

I plan to do more examples witheigshowin my next blog posting.

获取MATLAB代码

Published with MATLAB® R2013b

Published with MATLAB® R2013b