Any matrix is the sum of a symmetric matrix and a skew symmetric matrix.
Any matrix may be written as the sum of an antisymmetric matrix and a symmetric matrix.
Theorem 6.14 Every symmetric matrix is equivalent to a diagonal matrix.
The eigenvalues and eigenvector of a (real) symmetric matrix are all real.
Recall that the eigenvalues of a real symmetric matrix are real.
Every matrix is the sum of a symmetric and an antisymmetric matrix.
Thus by construction is always a real symmetric matrix.
for P, which is a positive definite symmetric matrix.
Show that every real matrix may be written as the sum of a skew symmetric and a symmetric matrix.
Similarly, the product AT A is a symmetric matrix.
In linear algebra and bilinear, a symmetrical matrix is a square matrix which equal to its characteristic is transposed.
because P is a symmetric positive-definite matrix.
Note that the covariance matrix is always a positive semi-definite matrix.
Note that $A^TA$ is a positive definite symplectic matrix.
The linear coefficients (Lij) of an equation system derived from the linear combinations form a symmetrical matrix.
Example: A diagonal matrix and its associated quadratic form.
A real symmetric $n \times n$ matrix $A$ is called positive definite if
All eigenvalues of a positive definite matrix are positive.
All positive integral powers of symmetric matrix is symmetric.
Inverse of the Hilbert matrix of size n. c is a real, symmetric matrix.
Express a matrix $A$ as the sum of a symmetric and a skew symmetric matrix
Then the principal curvatures are the eigenvalues of the symmetric matrix
Since is a symmetric real matrix, it can be written in the form:
Since the middle matrix is symmetric, it can be diagonalized.
The 3 x 3 symmetric matrix is called the effective reciprocal inertia tensor.
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