Square Of A Symmetric Matrix

Symmetric matrices appear in many different contexts. In statistics the covariance matrix is an example of a symmetric matrix. In engineering the so-called elastic strain matrix and the moment of inertia tensor provide examples. The crucial thing about symmetric matrices is stated in the main theorem of this section.

for all indices and .. Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative.. In linear algebra, a real symmetric matrix represents a self-adjoint operator 1 represented in an orthonormal basis over a real inner product space.

A matrix that is congruent with a symmetric matrix must also be symmetric. If a symmetric matrix is invertible, then its inverse matrix is also symmetric. The same happens with the adjoint of a symmetric matrix the adjoint matrix of a symmetric matrix results in another symmetric matrix. A symmetric matrix with real values is also a normal matrix.

theory as adjacency matrices etc. etc. Symmetric matrices play the same role as real numbers do among the complex numbers. Their eigenvalues often have physical or geometrical interpretations. One can also calculate with symmetric matrices like with numbers for example, we can solve B2 A for B if A is symmetric matrix and B is square root of A.

7 Symmetric Matrices 7.1 Diagonalization of Symmetric Matrices A symmetric matrix is a matrix Asuch that AT A. Such a matrix is necessarily square. Its main diagonal entries are arbitrary, but its other entries occur in pairs, on opposite sides of the main diagonal. Example 1. Determine which matrix is symmetric. a 3 5 5 3 b 2 4 0 8 3 8 0 4

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Symmetric Matrix. A symmetric matrix is a square matrix that is equal to its transpose. Mathematically, a matrix 92 A 92 is symmetric if 92 A AT 92 where 92 AT 92 is the transpose of 92 A 92, meaning that the rows and columns of 92 A 92 are interchanged. Examples of Symmetric Matrices Example 1 A 22 Symmetric Matrix. Consider the matrix

A symmetric matrix in linear algebra is a square matrix that remains unaltered when its transpose is calculated. That means, a matrix whose transpose is equal to the matrix itself, is called a symmetric matrix. It is mathematically defined as follows A square matrix B which of size n n is considered to be symmetric if and only if B T B. Consider the given matrix B, that is, a square

Definition of a Symmetric Matrix. A square matrix A is symmetric if and only if A A T where A T is the transpose of matrix A . A symmetric matrix may be recognized visually The entries that are symmetrically positioned with respect to the main diagonal are equal as shown in the example below of a symmetric matrix. These are examples of

A symmetric matrix is a square matrix that satisfies ATA, 1 where AT denotes the transpose, so a_ija_ji. This also implies A-1ATI, 2 where I is the identity matrix. For example, A4 1 1 -2 3 is a symmetric matrix. Hermitian matrices are a useful generalization of symmetric matrices for complex matrices. A matrix that is not symmetric is said to be an asymmetric