3 added 66 characters in body

This is quite an interesting question, perhaps a research problem. I think an elementary answer should be a high school algebra answer in the sense of Abhyankar and it would have to be in the spirit of what follows. But first a little story.

I was teaching linear algebra and had just covered eigenvalues and characteristic polynomials but was not yet at the chapter on the spectral theorem for real symmetric matrices. I was looking for problems to assign for my students as homework in the textbook we were using. One of the exercises was to show that a real matrix $$A=\left[ \begin{array}{cc} \alpha & \beta \\ \beta & \gamma \end{array} \right]$$ only had real eigenvalues. Not too hard. Write the characteristic polynomial $$\chi(\lambda)=det(\lambda I-A)=\lambda^2-(\alpha+\gamma)\lambda+\alpha\gamma-\beta^2$$ then its discriminant is $$\Delta=(\alpha+\gamma)^2-4(\alpha\gamma-\beta^2)=(\alpha+\gamma)^2+4\beta^2\ge 0\ .$$ Hence two real roots.

The next problem in the book was to do the same for $$A=\left[ \begin{array}{ccc} \alpha & \beta & \gamma\\ \beta & \delta & \varepsilon \\ \gamma & \varepsilon & \zeta \end{array} \right]$$ and (silly me) I also assigned it...

Here is the solution in the 3X3 case. All roots are real if the discriminant (for a binary cubic) is nonnegative. The discriminant of the characteristic polynomial is $$\Delta = (\delta \varepsilon ^{2} + \delta \zeta ^{2} - \zeta \delta ^{2} - \zeta \varepsilon ^{2} + \zeta \alpha ^{2} + \zeta \gamma ^{2} - \alpha \gamma ^{2} - \alpha \zeta ^{2} + \alpha \beta ^{2} + \alpha \delta ^{2} - \delta \alpha ^{2} - \delta \beta ^{2})^{2} \\ \mbox{} + 14(\delta \gamma \varepsilon - \beta \varepsilon ^{2} + \beta \gamma ^{2} - \alpha \gamma \varepsilon )^{2} \\ \mbox{} + 2(\delta \alpha \gamma + \delta \beta \varepsilon + \delta \gamma \zeta - \gamma \delta ^{2} - \gamma \varepsilon ^{2} + \gamma ^{3} - \alpha \beta \varepsilon - \alpha \gamma \zeta )^{2} \\ \mbox{} + 2(\delta \beta \gamma + \delta \varepsilon \zeta - \varepsilon ^{3} + \varepsilon \alpha ^{2} + \varepsilon \gamma ^{2} - \alpha \beta \gamma - \alpha \delta \varepsilon - \alpha \varepsilon \zeta )^{2} \\ \mbox{} + 2(\zeta \alpha \beta + \zeta \beta \delta + \zeta \gamma \varepsilon - \beta \varepsilon ^{2} - \beta \zeta ^{2} + \beta ^{3} - \delta \alpha \beta - \alpha \gamma \varepsilon )^{2} \\ \mbox{} + 14(\zeta \beta \varepsilon - \gamma \varepsilon ^{2} + \gamma \beta ^{2} - \alpha \beta \varepsilon )^{2} \\ \mbox{} + 2(\zeta \beta \gamma + \delta \varepsilon \zeta - \varepsilon ^{3} + \varepsilon \alpha ^{2} + \varepsilon \beta ^{2} - \alpha \beta \gamma - \alpha \delta \varepsilon - \alpha \varepsilon \zeta )^{2} \\ \mbox{} + 14(\varepsilon \beta ^{2} + \zeta \beta \gamma - \delta \beta \gamma - \varepsilon \gamma ^{2})^{2} \\ \mbox{} + 2(\zeta \alpha \beta + \zeta \beta \delta + \zeta \gamma \varepsilon - \beta \gamma ^{2} - \beta \zeta ^{2} + \beta ^{3} - \delta \alpha \beta - \delta \gamma \varepsilon )^{2} \\ \mbox{} + 2(\alpha \gamma \zeta + \zeta \beta \varepsilon - \gamma ^{3} + \gamma \beta ^{2} + \gamma \delta ^{2} - \delta \alpha \gamma - \delta \beta \varepsilon - \delta \gamma \zeta )^{2}\ .$$

This formula comes from a paper by Ilyushechkin in Mat. Zametki, 51, 16-23, 1992.

I suspect the elementary answer should be as follows. First find a list of invariants or covariants of binary forms $C_1,C_2,\ldots$ such that a form with real coefficients has only real roots iff these covariants are nonnegative. Apply this to the characteristic polynomial of a general real symmetric matrix and show that you get sums of squares. I suppose these covariants, via Sturm's sequence type arguments, should correspond to subresultants or rather subdiscriminants. This seems also related to Part 2) of Godsil's answer.

2 added 5 characters in body

This is quite an interesting question, perhaps a research problem. I think an elementary answer should be a high school algebra answer in the sense of Abhyankar and it would have to be in the spirit of what follows. But first a little story.

I was teaching linear algebra and had just covered eigenvalues and characteristic polynomials but was not yet at the chapter on the spectral theorem for real symmetric matrices. I was looking for problems to assign for my students as homework in the textbook we were using. One of the exercises was to show that a real matrix $$A=\left[ \begin{array}{cc} \alpha & \beta \\ \beta & \gamma \end{array} \right]$$ only had real eigenvalues. Not too hard. Write the characteristic polynomial $$\chi(\lambda)=det(\lambda I-A)=\lambda^2-(\alpha+\gamma)\lambda+\alpha\gamma-\beta^2$$ then its discriminant is $$\Delta=(\alpha+\gamma)^2-4(\alpha\gamma-\beta^2)=(\alpha+\gamma)^2+4\beta^2\ge 0\ .$$ Hence two real roots.

The next problem in the book was to do the same for $$A=\left[ \begin{array}{ccc} \alpha & \beta & \gamma\\ \beta & \delta & \varepsilon \\ \gamma & \varepsilon & \zeta \end{array} \right]$$ and (silly me) I also assigned it...

Here is the solution in the 3X3 case. All roots are real if the discriminant (for a binary cubic) is nonnegative. The discriminant of the characteristic polynomial is $$\Delta = (\delta \varepsilon ^{2} + \delta \zeta ^{2} - \zeta \delta ^{2} - \zeta \varepsilon ^{2} + \zeta \alpha ^{2} + \zeta \gamma ^{2} - \alpha \gamma ^{2} - \alpha \zeta ^{2} + \alpha \beta ^{2} + \alpha \delta ^{2} - \delta \alpha ^{2} - \delta \beta ^{2})^{2} \\ \mbox{} + 14(\delta \gamma \varepsilon - \beta \varepsilon ^{2} + \beta \gamma ^{2} - \alpha \gamma \varepsilon )^{2} \\ \mbox{} + 2(\delta \alpha \gamma + \delta \beta \varepsilon + \delta \gamma \zeta - \gamma \delta ^{2} - \gamma \varepsilon ^{2} + \gamma ^{3} - \alpha \beta \varepsilon - \alpha \gamma \zeta )^{2} \\ \mbox{} + 2(\delta \beta \gamma + \delta \varepsilon \zeta - \varepsilon ^{3} + \varepsilon \alpha ^{2} + \varepsilon \gamma ^{2} - \alpha \beta \gamma - \alpha \delta \varepsilon - \alpha \varepsilon \zeta )^{2} \\ \mbox{} + 2(\zeta \alpha \beta + \zeta \beta \delta + \zeta \gamma \varepsilon - \beta \varepsilon ^{2} - \beta \zeta ^{2} + \beta ^{3} - \delta \alpha \beta - \alpha \gamma \varepsilon )^{2} \\ \mbox{} + 14(\zeta \beta \varepsilon - \gamma \varepsilon ^{2} + \gamma \beta ^{2} - \alpha \beta \varepsilon )^{2} \\ \mbox{} + 2(\zeta \beta \gamma + \delta \varepsilon \zeta - \varepsilon ^{3} + \varepsilon \alpha ^{2} + \varepsilon \beta ^{2} - \alpha \beta \gamma - \alpha \delta \varepsilon - \alpha \varepsilon \zeta )^{2} \\ \mbox{} + 14(\varepsilon \beta ^{2} + \zeta \beta \gamma - \delta \beta \gamma - \varepsilon \gamma ^{2})^{2} \\ \mbox{} + 2(\zeta \alpha \beta + \zeta \beta \delta + \zeta \gamma \varepsilon - \beta \gamma ^{2} - \beta \zeta ^{2} + \beta ^{3} - \delta \alpha \beta - \delta \gamma \varepsilon )^{2} \\ \mbox{} + 2(\alpha \gamma \zeta + \zeta \beta \varepsilon - \gamma ^{3} + \gamma \beta ^{2} + \gamma \delta ^{2} - \delta \alpha \gamma - \delta \beta \varepsilon - \delta \gamma \zeta )^{2}\ .$$

This formula comes from a paper by Ilyushechkin in Mat. Zametki, 51, 16-23, 1992.

I suspect the elementary answer should be as follows. First find a list of invariants or covariants of binary forms $C_1,C_2,\ldots$ such that a form with real coefficients has only real roots iff these covariants are nonnegative. Apply this to the characteristic polynomial of a general real syymetric symmetric matrix and show that you get sums of squares. I suppose these covariants, via Sturm's sequence type arguments, should correspond to subresultants or rather subdiscriminants. This seems also related to Part 2) of Godsil's answer.

1

This is quite an interesting question, perhaps a research problem. I think an elementary answer should be a high school algebra answer in the sense of Abhyankar and it would have to be in the spirit of what follows. But first a little story.

I was teaching linear algebra and had just covered eigenvalues and characteristic polynomials but was not yet at the chapter on the spectral theorem for real symmetric matrices. I was looking for problems to assign for my students as homework in the textbook we were using. One of the exercises was to show that a real matrix $$A=\left[ \begin{array}{cc} \alpha & \beta \\ \beta & \gamma \end{array} \right]$$ only had real eigenvalues. Not too hard. Write the characteristic polynomial $$\chi(\lambda)=det(\lambda I-A)=\lambda^2-(\alpha+\gamma)\lambda+\alpha\gamma-\beta^2$$ then its discriminant is $$\Delta=(\alpha+\gamma)^2-4(\alpha\gamma-\beta^2)=(\alpha+\gamma)^2+4\beta^2\ge 0\ .$$ Hence two real roots.

The next problem in the book was to do the same for $$A=\left[ \begin{array}{ccc} \alpha & \beta & \gamma\\ \beta & \delta & \varepsilon \\ \gamma & \varepsilon & \zeta \end{array} \right]$$ and (silly me) I also assigned it...

Here is the solution in the 3X3 case. All roots are real if the discriminant (for a binary cubic) is nonnegative. The discriminant of the characteristic polynomial is $$\Delta = (\delta \varepsilon ^{2} + \delta \zeta ^{2} - \zeta \delta ^{2} - \zeta \varepsilon ^{2} + \zeta \alpha ^{2} + \zeta \gamma ^{2} - \alpha \gamma ^{2} - \alpha \zeta ^{2} + \alpha \beta ^{2} + \alpha \delta ^{2} - \delta \alpha ^{2} - \delta \beta ^{2})^{2} \\ \mbox{} + 14(\delta \gamma \varepsilon - \beta \varepsilon ^{2} + \beta \gamma ^{2} - \alpha \gamma \varepsilon )^{2} \\ \mbox{} + 2(\delta \alpha \gamma + \delta \beta \varepsilon + \delta \gamma \zeta - \gamma \delta ^{2} - \gamma \varepsilon ^{2} + \gamma ^{3} - \alpha \beta \varepsilon - \alpha \gamma \zeta )^{2} \\ \mbox{} + 2(\delta \beta \gamma + \delta \varepsilon \zeta - \varepsilon ^{3} + \varepsilon \alpha ^{2} + \varepsilon \gamma ^{2} - \alpha \beta \gamma - \alpha \delta \varepsilon - \alpha \varepsilon \zeta )^{2} \\ \mbox{} + 2(\zeta \alpha \beta + \zeta \beta \delta + \zeta \gamma \varepsilon - \beta \varepsilon ^{2} - \beta \zeta ^{2} + \beta ^{3} - \delta \alpha \beta - \alpha \gamma \varepsilon )^{2} \\ \mbox{} + 14(\zeta \beta \varepsilon - \gamma \varepsilon ^{2} + \gamma \beta ^{2} - \alpha \beta \varepsilon )^{2} \\ \mbox{} + 2(\zeta \beta \gamma + \delta \varepsilon \zeta - \varepsilon ^{3} + \varepsilon \alpha ^{2} + \varepsilon \beta ^{2} - \alpha \beta \gamma - \alpha \delta \varepsilon - \alpha \varepsilon \zeta )^{2} \\ \mbox{} + 14(\varepsilon \beta ^{2} + \zeta \beta \gamma - \delta \beta \gamma - \varepsilon \gamma ^{2})^{2} \\ \mbox{} + 2(\zeta \alpha \beta + \zeta \beta \delta + \zeta \gamma \varepsilon - \beta \gamma ^{2} - \beta \zeta ^{2} + \beta ^{3} - \delta \alpha \beta - \delta \gamma \varepsilon )^{2} \\ \mbox{} + 2(\alpha \gamma \zeta + \zeta \beta \varepsilon - \gamma ^{3} + \gamma \beta ^{2} + \gamma \delta ^{2} - \delta \alpha \gamma - \delta \beta \varepsilon - \delta \gamma \zeta )^{2}\ .$$

This formula comes from a paper by Ilyushechkin in Mat. Zametki, 51, 16-23, 1992.

I suspect the elementary answer should be as follows. First find a list of invariants or covariants of binary forms $C_1,C_2,\ldots$ such that a form with real coefficients has only real roots iff these covariants are nonnegative. Apply this to the characteristic polynomial of a general real syymetric matrix and show you get sums of squares. I suppose these covariants, via Sturm's sequence type arguments, should correspond to subresultants or rather subdiscriminants. This seems also related to Part 2) of Godsil's answer.