Symmetric Matrices

We all know that Matrix is an ordered form of easy presentation of data in an array of rows and columns. There are some special

types of matrices. One of them is "Symmetric Matrix". Here, we study about Symmetric Matrices, broad outlines.

What is a symmetric matrix?

Two essentital things are required for a symmetric matrix.

1.  It should be a square matrix.  i.e. No. of rows in the matrix = No. of columns in the matrix

2. It is equal to its Transporse Matrix.  A= A^T.

So if these two conditions are satisfied, the matrix is symmetric.  The entries of the matrix are symmetric about the main diagonal of the matrix.

Eg of a symmetric matrix  =

Matrix(A)   In other words, a ij = a ji for all i and j in a symmetric matrix.

The diagonal matric    1 0 0   is also symmetric as here also a ij = a ji for all i not equal to j
and a ij = 0 for i not equal to j
0 1 0
0 0 1


When we talk about symmetric, the question of orthogonal matrix also comes.

What is an orthogonal Matrix?

A matrix P is called orthogonal if its columns form an orthonormal set and call a matrix A orthogonally diagonalisable if it can be

diagonalized by D =P inverse AP with P an orthogonal Matrix.

Important properties of symmetric matrices

Consider the following symmetric matrices.

1      2    3                                and                                            2     3      4

2      4    5                                                                                 3     4      5

3      5    6                                                                                 4       5     6

Let them be called A  and B.

Now consider A+B    =      3   5    7

5   8    10

7  10    12

It is symmetric.  So addition of two symmetric matrices lead to another symmetric matrix.

Subtraction:           A-B     =      -1    -1    -1

-1     0      0

-1     0     0

Again a symmetric matrix.  So addition and subtraction of two symmetric matrices preserve the symmetry property.

Multiplication:       Consider A*B   =        Again a symmetric matrix

Skew-symmetric matrix:   A skew symmetric matrix is one whose transpose = negative of the matrix

In other words, A transpose   =   -A

Here the condition is a ij =-a ji

Eigenvalues and eigenvectors of a symmetric matrix are real:

The eigenvalues of a symmetric matrix are real.   Just because the coefficients of a matrix are real certainly does not imply eigen

values are real.  But for symmetric matrix, eigen values are always real.

Decomposition of a symmetric matrices

Every square real matrix can be written as a product of two real symmetric matrices, and every square complex matrix can be written as a product of two complex symmetric matrices. Here, Jordan's normal form is used.

Every real non-singular matrix can be uniquely factored as the product of an orthogonal matrix and a symmetric Positive definite matrix.  This process we call as polar decomposition.  But for singular matrices, they can be factored but not uniquely.

Cholesky decomposition states that every real positive-definite symmetric matrix is a product of an upper-triangular matrix and its transpose.

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Every real symmetric matrix A can be diagonalized, moreover the eigen decomposition takes a simpler form:

Image(X)

where Q' is an orthogonal matrix (the columns of which are eigenvectors of A), and Λ is real and diagonal (having the eigenvalues of A on the diagonal.
Determinant
Determinant
A linear transformation on R2 given by the indicated matrix. The determinant of this matrix is −1, as the area of the green parallelogram at the right is 1, but the map reverses the orientation, since it turns the counterclockwise orientation of the vectors to a clockwise one.

The determinant det(A) or |A| of a square matrix A is a number encoding certain properties of the matrix. A matrix is invertible if and only if its determinant is nonzero. Its absolute value equals the area (in R2) or volume (in R3) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved.

The determinant of 2-by-2 matrices is given by

Determinant(A)
Because a symmetric matrix is a square , a determinant exists for every symmetric matrix.