28 QUADRATIC FUNCTIONS AND SYMMETRIC MATRICES
28.1. Symmetric matrices and quadratic functions
Symmetric matrices
A square matrix is symmetric if it is equal to its transpose. That is,
The set of symmetric matrices is denoted . This set is a subspace of .
Example 1: A symmetric matrix. |
The matrix |
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is symmetric. The matrix |
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is not, since it is not equal to its transpose.
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See also:
- Representation of a weighted, undirected graph.
- Laplacian matrix of a graph.
- Hessian of a function.
- Gram matrix of data points.
Quadratic functions
A function is said to be a quadratic function if it can be expressed as
for numbers , , and , . A quadratic function is thus an affine combination of the ‘s and all the ‘‘cross-products’’ . We observe that the coefficient of is .
The function is said to be a quadratic form if there are no linear or constant terms in it:
Note that the Hessian (matrix of second-derivatives) of a quadratic function is constant.
Examples:
Link between quadratic functions and symmetric matrices
There is a natural relationship between symmetric matrices and quadratic functions. Indeed, any quadratic function can be written as
for an appropriate symmetric matrix , vector and scalar . Here:
- is the coefficient of in ;
- for , is the coefficient of the term in ;
- is the coefficient of the term ;
- is the constant term, .
If is a quadratic form, then , , and we can write where .
Examples: Two-dimensional example.
28.2. Second-order approximations of non-quadratic functions
We have seen how linear functions arise when one seeks a simple, linear approximation to a more complicated non-linear function. Likewise, quadratic functions arise naturally when one seeks to approximate a given non-quadratic function by a quadratic one.
One-dimensional case
If is a twice-differentiable function of a single variable, then the second-order approximation (or, second-order Taylor expansion) of at a point is of the form
where is the first derivative, and the second derivative, of at . We observe that the quadratic approximation has the same value, derivative, and second-derivative as , at .
Example 2: The figure shows a second-order approximation of the univariate function , with values | |
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at the point (in red). | |
Multi-dimensional case
In multiple dimensions, we have a similar result. Let us approximate a twice-differentiable function by a quadratic function , so that and coincide up and including to the second derivatives.
The function must be of the form
where , , and . Our condition that coincides with up and including to the second derivatives shows that we must have
where is the Hessian, and the gradient, of at .
Solving for we obtain the following result:
Second-order expansion of a function. The second-order approximation of a twice-differentiable function at a point is of the form
where is the gradient of at , and the symmetric matrix is the Hessian of at .
Example: Second-order expansion of the log-sum-exp function.
28.3. Special symmetric matrices
Diagonal matrices
Perhaps the simplest special case of symmetric matrices is the class of diagonal matrices, which are non-zero only on their diagonal.
If , we denote by , or for short, the (symmetric) diagonal matrix with on its diagonal. Diagonal matrices correspond to quadratic functions of the form
Such functions do not have any ‘‘cross-terms’’ of the form with .
Example 3: A diagonal matrix and its associated quadratic form. |
Define a diagonal matrix: |
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For the matrix above, the associated quadratic form is |
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Symmetric dyads
Another important class of symmetric matrices is that of the form , where . The matrix has elements and is symmetric. Such matrices are called symmetric dyads. (If , then the dyad is said to be normalized.)
Symmetric dyads correspond to quadratic functions that are simply squared linear forms: .
Example: A squared linear form.