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Trying to diagonalize the matrix:

$$A=\begin{pmatrix}2 & -\frac{1}{2} & -\frac{1}{2} \\-\frac{1}{2} & 3 & -\frac{1} {2} \\-\frac{1}{2} & -\frac{1}{2} & 5\end{pmatrix},$$

I got the characteristic polynomial :

$$p(\lambda)=\lambda^3-10\lambda^2+\frac{121}{4}\lambda-30.$$

Is it even possible to solve for $\lambda$?

Servaes
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Charlene
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  • Looks like finding a diagonalization in closed form might be messy... – Surb Aug 07 '15 at 23:12
  • What book are you using, please? I have been seeing this sort of problem here lately, I usually answer with Hermite's method as shown in books about integer quadratic forms ( such as yours). At least one book is teaching a cookbook method, see http://math.stackexchange.com/questions/1382288/finding-p-such-that-ptap-is-a-diagonal-matrix – Will Jagy Aug 08 '15 at 00:59
  • I see, this is also yours: http://math.stackexchange.com/questions/1388281/find-the-transitional-matrix-that-would-transform-this-form-to-a-diagonal-form – Will Jagy Aug 08 '15 at 01:48
  • Diagonalise from first principles. – Allawonder Apr 11 '20 at 13:03

3 Answers3

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It seems you've miscalculated the characteristic polynomial; it should be $$\lambda^3-10\lambda^2+\tfrac{121}{4}\lambda-\tfrac{109}{4}.$$ I see no nice way to find the roots other than brute force...

Servaes
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  • there is a separate procedure that does not use eigenvalues; it does not *orthogonally* diagonalize anything, in fact it just finds a matrix $P$ with determinant $1$ and $P^t A P = D$ variant of Hermite's method and more cookbook, see http://math.stackexchange.com/questions/1382288/finding-p-such-that-ptap-is-a-diagonal-matrix – Will Jagy Aug 08 '15 at 01:02
  • well, please see http://math.stackexchange.com/questions/1388281/find-the-transitional-matrix-that-would-transform-this-form-to-a-diagonal-form about the same matrix, by the same person, and my new http://math.stackexchange.com/questions/1388421/reference-for-linear-algebra-books-that-teach-reverse-hermite-method-for-symmetr – Will Jagy Aug 08 '15 at 01:51
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There are actual formulas for finding the roots of cubic (and even quartic) equations in terms of the coefficients, which you can look up on the Web. As a practical matter, you might try graphing the polynomial and looking to see if there is at least one root you can guess from the graph. If there is, then you can factor that out of the equation and get a quadratic equation for the remaining two roots that you know how to solve.

user255896
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I think you need to find eigenvalues of the matrix and put it on diagonal elements by sorting from largest to smallest.

you can use QR algorithm for finding the eigenvalues.

at the end by performing the QR algorithm, you will get the diagonally aligned matrix, what you were needing.

$$A(0) = QR;$$

$$A(t) = RQ;$$

$$A(t + 1) = RQ(t);$$

the algorithm converges to diagonal matrix, consisting from eigenvalues.

in this context Q is the matrix containing orhonormal vectors and R is the upper triangular matrix

Chris Tang
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Grigor Nazaryan
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