Lay Chapter 7,
Symmetric Matrices and Quadratic Forms
Diagonalize a Matrix
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Calculate the eigenvalues of a.
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Calculate the eigenvectors of a.
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Normalize the columns of v
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Check.
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Diagonalize a Matrix
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Calculate the eigenvalues of a.
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Calculate the eigenvectors of a.
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Normalize the columns of v
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Check.
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Matrix Representation of a Quadratic Form
Symmetric matrices represent quadratic forms in the following way.
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Here is the quadratic form generated by a diagonal matrix
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... and another generated by a symmetric matrix which is not a diagonal matrix.
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The difference is the "mixed" term.
Matrix Representation of Quadratic Forms
Write the matrix representing a particular quadratic form.
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Value of a Quadratic Form at a Point
Evaluate a particular quadratic form at distinct points.
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Change of Variable in a Quadratic Form
If x is a variable in , thena change of variable in a quadratic form is an equation of the form.
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where y is a variable in and p is an invertible n x n matrix.
Diagonalize the following quadratic form.
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First, calculate the corresponding diagonal matrix.
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Then the columns of the matrix p are the orthonormal eigenvectors of a, in the proper order.
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Check the diagonalization.
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Check that the two quadratic forms agree for x = {2,-2}.
First, here is qx applied to x = {2,-2}.
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Now x = p.y implies that y = Inverse[p].x
Here is qy applied to y = Inverse[p].x
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Colorful Vectors and Curves
Contains Mathematica Code Only -- No Solved Exercises
Principal Axes
Let's recast a quadratic form generated by a symmetric 2 x 2 matrix as a function of two variables.
We use the matrix a from the previous example.
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Now do the same thing for the matrix d.
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The second surface s obtained from the first by a rotation.
The y1- and y2-axes in the second figure correspond to the principal axes in the first.
The principal axes are the directions indicated by the columns of the matrix p used in describing the change of variables: x = p y.
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Let's see those two vectors. We will exaggerate their lengths in order to see them better.
Note the orientation of the standard coordinate axes.
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In this image as well, the lengths of the vectors indicating the principal axes are exagerated in order to see them better.
One of the vectors is mostly underneath the surface.
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Using Contour Plots to Investigate Quadratic Forms and their Principal Axes
Let's see a contour plot of the quadratic form qa.
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... and a contour plot of the quadratic form qd.
It seems clear that the second is obtained by a rotation of the first.
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Now let's see those principal axes superimposed on the contour plot of qa.
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Classification of Quadratic Forms
Positive definite
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Negative definite
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Indefinite
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Positive Semidefinite
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Diagonalization a Quadratic Form
Diagonalize the following quadratic form.
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First, calculate the corresponding diagonal matrix.
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Use d to construct the equivalent (diagonal) quadratic form.
This quadratic form is Lay 7.2.17.
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Now relate the two quadratic forms by calculating the change of basis matrix p such that x = p y.
The columns of the matrix p are the orthonormal eigenvectors of a, in the proper order.
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Check the diagonalization.
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Constrained Optimization
Consider the following quadratic form:
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Calculate the corresponding diagonal matrix.
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The columns of the matrix p are the orthonormal eigenvectors of a, taken in the proper order.
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Check the diagonalization.
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Constrained Optimization
Consider the following quadratic form.
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Calculate the corresponding diagonal matrix.
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Then the columns of the matrix p are the orthonormal eigenvectors of a, taken in the proper order.
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Check the diagonalization.
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We can graph this surface.
Stretch out the graph for a better view.
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Created by Mathematica (December 11, 2004) | ![]() |