Show that for any square matrix, the matrix A +(A)^t is sym

In summary, a square matrix A added to its transpose (A)^t results in a symmetric matrix, which means that the entries above the diagonal are equal to those below the diagonal. This is because the transpose of a matrix involves interchanging the rows and columns, making the matrix symmetric with respect to the main diagonal.
  • #1
MidgetDwarf
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Show that for any square matrix, the matrix A + ( A )^t is symmetric.

My attempt. I know that A square matrix has the property that asub (ij). Where i=1,..., m and j=1,..,n.
M=n(same number of rows and columns).

I know that a transpose of a matrix means to interchange the rows with columns.

What I do not understand what it means for a matrix to be symmetric?
 
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  • #2
Sorry I forgot the t means the transpose if anyone is confused about my notation.
 
  • #3
MidgetDwarf said:
What I do not understand what it means for a matrix to be symmetric?
A matrix B is symmetric if BT=B

In other words Bmn=Bnm for all entries (which also means it must be square). The diagonal entries can be anything because, for example, B11=B11 is always true. But B12 must equal B21, and so on.

(It's easy to understand visually.)
 

1. What is the definition of a symmetric matrix?

A symmetric matrix is a square matrix that is equal to its transpose, meaning that the elements on the diagonal are the same and the elements above and below the diagonal are reflections of each other.

2. How do you show that the matrix A +(A)^t is sym?

To show that a matrix is symmetrical, we need to prove that it is equal to its own transpose. In this case, we can expand the expression (A + (A)^t) to get (A + (A)^t)^t, and using the fact that the transpose of a sum is equal to the sum of the transposes, we get (A)^t + ((A)^t)^t, which simplifies to A + A, or simply 2A. Since A is equal to its own transpose, 2A is symmetrical.

3. What is the significance of a symmetric matrix?

Symmetric matrices have many important properties and applications in mathematics and science. For example, they can be used to represent quadratic forms, which are used in optimization problems. They also have special properties that make them easier to solve and manipulate in certain situations.

4. Can a non-square matrix be symmetrical?

No, a non-square matrix cannot be symmetrical because it is not possible for a matrix to be equal to its own transpose unless it is a square matrix.

5. How does the symmetry of a matrix affect its eigenvalues and eigenvectors?

A symmetric matrix has real eigenvalues and orthogonal eigenvectors. This means that the eigenvalues are all real numbers and the corresponding eigenvectors are perpendicular to each other. This makes symmetric matrices useful in solving eigenvalue problems and diagonalizing matrices.

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