Is xTAx always non-zero for a real, symmetric, nonsingular matrix A?

Click For Summary
For a real, symmetric, nonsingular matrix A, the expression xTAx is indeed non-zero for any non-zero vector x. This is because such matrices are diagonalizable, and in their diagonal form, the eigenvalues are all positive due to the nonsingularity of A. Consequently, the quadratic form xTAx will yield a positive value for any non-zero x. The discussion emphasizes the importance of diagonalization in understanding the properties of the matrix. Therefore, xTAx is always non-zero for x≠0.
elimax
Messages
2
Reaction score
0
Basic question, I think, but I'm not sure. It is a step in a demonstration, so it would be nice if it were true.
smile.gif


True or false? Why? If A is a real, symmetric, nonsingular matrix, then xTAx≠0 for x≠0.
 
Physics news on Phys.org
Since ##A## is real and symmetric, it is diagonalizable. So you may start by thinking about ##x^T A x## in a basis in which ##A## is diagonal.
 
Thread 'How to define a vector field?'

Thread 'How to define a vector field?'

Hello! In one book I saw that function ##V## of 3 variables ##V_x, V_y, V_z## (vector field in 3D) can be decomposed in a Taylor series without higher-order terms (partial derivative of second power and higher) at point ##(0,0,0)## such way: I think so: higher-order terms can be neglected because partial derivative of second power and higher are equal to 0. Is this true? And how to define vector field correctly for this case? (In the book I found nothing and my attempt was wrong...
View full post »

Similar threads

Undergrad Congruence for Symmetric and non-Symmetric Matrices for Quadratic Form
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Is a symmetric matrix with positive eigenvalues always real?
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Canonical Form for quadratic equations *with* linear terms
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Quintic Polynomial Tschirnhaus Transform: Possible complex Bring Radicals
  • · Replies 0 ·
Replies
0
Views
2K
MHB Zero-Trace Symmetric Matrix is Orthogonally Similar to A Zero-Diagonal Matrix.
  • · Replies 5 ·
Replies
5
Views
5K
Undergrad General form of symmetric 3x3 matrix with only 2 eigenvalues
  • · Replies 4 ·
Replies
4
Views
6K
Undergrad Uniqueness of reduced row echelon form (proof verification)
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Meaning of Third Eigenvalue in a Tilted Ellipse in a 3x3 Matrix
  • · Replies 3 ·
Replies
3
Views
2K
Determinant and symmetric positive definite matrix
  • · Replies 1 ·
Replies
1
Views
2K
MHB Using matrix to complete the square
  • · Replies 4 ·
Replies
4
Views
3K