If A Matrix A Is Symmetric As Well As Skew Symmetric

09 May, 2021

The proof can be made without referring to eigenvalues - it is possible to prove the claim only with the application of basic definitions for symmetric matrix and nilpotent matrix Note that for A which is symmetric matrix A 2 T A A T A T A T A T 2 A 2 so square of symmetric matrix is also symmetric. A j i 0 i j.

Example 22 Express Matrix B As Sum Of Symmetric And Skew

Check Answer a Tardigrade.

If a matrix a is symmetric as well as skew symmetric. For all i and j. If a matrix A is symmetric as well as skew symmetric then A is a A diagonal matrix B null matrix C unit matrix D none of these. Lets take an example of a matrix.

To ask your doubts on this topic and much more click here. A matrix can be skew symmetric only if it is square. A is a diagonal matrix B.

Consider a matrix A is skew-symmetric then A T A and A is symmetric then A T A Calculation. Since A is skew-symmetric. Hence option 4 is correct.

A square matrix is a matrix with the same number of rows and columns. Hence a i j 0 i. A T A Since A is symmetric.

If B is a square matrix and A is any square matrix of order equl to that of B prove that B AB is symmetirc or skew symmetric according as A is symmetric or skew symmetric. A T A A A 2A O A O Hence A is a null matrix. If the transpose of a matrix is equal to the negative of itself the matrix is said to be skew symmetric.

Since AT - A A is a skew - symmetric matrix. Take a 2 2 null matrix. Prove that A A T is a symmetric and A A T is a skew symmetric matrix where.

Since A is skew-symmetric. Prove that every Hermitian matrix A can be written as the sum ABiC where B is a real symmetric matrix and C is a real skew-symmetric matrix. Square Matrix A is said to be skew-symmetric if.

A is a zero matrix C. A is a square matrix D. In other words we can say that matrix A is said to be skew-symmetric if transpose of matrix A is equal to negative of Matrix.

An n-by-n matrix is known as a square matrix of order n. If matrix A is symmetricAT AIf matrix A is skew symmetricAT AAlso diagonal elements are zeroNow it is given that a matrix A is both symmetric as well as skew symmetric A AT Awhich is only possible if A is zero matrixA 0 0 0 0 AT ATherefore option B is correct answer. A 1 2 43 2 1-2 -3 2.

In other words we can say that matrix A is said to be skew-symmetric if transpose of matrix A is equal to negative of matrix A ie AT A. So B is the correct answer. None of these Since A is both symmetric and skew-symmetric matrix A A and A A Comparing both equations A A A A O 2A O A O Therefore A is a zero matrix.

Thus A 0 0 0 0 is an example of a matrix that is both symmetric and skew - symmetric. We need to find a square matrix which is both symmetric as well as skew symmetric. Symmetric and Skew Symmetric Matrices.

Note that all the main diagonal elements in the skew-symmetric matrix are zero. Show that any square matrix can be expressed as the sum of two matrices one symmetric and the other skew-symmetric. The video covers SYMMETRIC SKEW SYMMETRIC AND ORTHOGONAL MATRIX.

Show that any square matrix can be expressed as the sum of two matrices one symmetric and the other skew-symmetric. This means that for a matrix to be skew symmetric A-A. Express a Hermitian Matrix as a Sum of Real Symmetric Matrix and a Real Skew-Symmetric Matrix Recall that a complex matrix is called Hermitian if AA where AbarAtrans.

Also for the matrix for all the values of i and j. A i i 0 i 1 2 n and a j i a j i i j Also A is symmetric so a j i a j i i and j. Let A a i j n x m.

Square matrix A is said to be skew-symmetric if aij aji for all i and j.

If A Matrix A Is Both Symmetric And Skew Symmetric Then