Quick linear algebra determinant proof.

[Kuma]

Homework Statement

If A is a square symmetric matrix nxn. Show that the determinant of A is the product of its eigenvalues.

Homework Equations

The Attempt at a Solution

From spectral decomp.

A = QλQ'
|A| = |QλQ'| = |QQ'λ| = |Q||Q'||λ| = |λ| = the product of its diagonals (eigenvalues).

The step I'm not 100% sure of is if I can interchange QλQ' to QQ'λ

 

[I like Serena]

Homework Statement

If A is a square symmetric matrix nxn. Show that the determinant of A is the product of its eigenvalues.

Homework Equations

The Attempt at a Solution

From spectral decomp.

A = QλQ'
|A| = |QλQ'| = |QQ'λ| = |Q||Q'||λ| = |λ| = the product of its diagonals (eigenvalues).

The step I'm not 100% sure of is if I can interchange QλQ' to QQ'λ

Hi Kuma!

No, you can't interchange QλQ' to QQ'λ.
This can be seen because then QQ'λ=Iλ=λ, but this would not match your original matrix A.

However, there is no need to interchange them.
You can take the same steps without interchanging them.

 

[Ray Vickson]

Homework Statement

If A is a square symmetric matrix nxn. Show that the determinant of A is the product of its eigenvalues.

Homework Equations

The Attempt at a Solution

From spectral decomp.

A = QλQ'
|A| = |QλQ'| = |QQ'λ| = |Q||Q'||λ| = |λ| = the product of its diagonals (eigenvalues).

The step I'm not 100% sure of is if I can interchange QλQ' to QQ'λ

The constant in the characteristic equation of A equals the determinant of A. How is that constant related to the eigenvalues?

RGV