Math Problem Statement
Let A = [3 5 4 2] What is the maximum value of f(x) = x^TAx subject to the constraint x^Tx = 1 ?
Solution
The problem you're asking involves maximizing the quadratic form , where is a matrix and is the constraint, which represents that is a unit vector.
Step 1: Interpreting the Matrix
The matrix is given as . This seems to represent a diagonal matrix of the form:
3 & 0 & 0 & 0 \\ 0 & 5 & 0 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 2 \end{pmatrix}$$ ### Step 2: Problem Setup You want to maximize: $$f(x) = x^T A x$$ where $$x = (x_1, x_2, x_3, x_4)^T$$ is a 4-dimensional vector. Under the constraint $$x^T x = 1$$, this means that: $$x_1^2 + x_2^2 + x_3^2 + x_4^2 = 1$$ ### Step 3: Eigenvalue Interpretation In this case, $$f(x) = x^T A x$$ can be maximized by considering the eigenvalues of matrix $$A$$. Since $$A$$ is diagonal, the eigenvalues are simply the entries on the diagonal, which are $$3, 5, 4, 2$$. The quadratic form $$x^T A x$$ is maximized when $$x$$ aligns with the eigenvector corresponding to the largest eigenvalue. The largest eigenvalue of $$A$$ is $$5$$. ### Step 4: Conclusion The maximum value of $$f(x) = x^T A x$$, subject to the constraint $$x^T x = 1$$, is the largest eigenvalue of $$A$$, which is: $$\boxed{5}$$ Would you like further explanation on eigenvalues or the quadratic form, or do you have any other questions? Here are 5 relative questions you could explore: 1. How do you compute eigenvalues for non-diagonal matrices? 2. What happens if $$A$$ is not diagonal in a quadratic form problem? 3. Can you provide a geometric interpretation of quadratic forms? 4. How would the maximization change if the constraint $$x^T x = 1$$ were different? 5. What role do orthogonal matrices play in simplifying quadratic forms? **Tip:** When dealing with quadratic forms, diagonal matrices provide a quick way to interpret maximum or minimum values through their eigenvalues.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Quadratic Forms
Eigenvalues
Formulas
f(x) = x^T A x
x^T x = 1
Eigenvalue equation: A v = λ v
Theorems
Spectral Theorem
Rayleigh Quotient
Suitable Grade Level
Undergraduate Linear Algebra
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