Here you will learn how to find trace of matrix, its properties and what is orthogonal matrix with example.
Let’s begin –
Trace of Matrix
The sum of the elements of the square matrix A lying along the principal diagonal is called the trace of A i.e (tr(A)).
Thus if A = \([a_{ij}]_{n\times n}\),
then tr(A) = \(\sum_{i=1}^{n}\) \(a_{ii}\) = \(a_{11}\) + \(a_{22}\) + ……… + \(a_{nn}\).
How to Find Trace of Matrix :
for example, for 3×3 matrix, if A = \(\begin{bmatrix} 2 & 1 & -1 \\ 3 & -2 & 5 \\ 1 & 5 & 3 \end{bmatrix}\)
then, trace of A or tr(A) = 2 + (-2) + 3 = 3
for example, for 2×2 matrix, if A = \(\begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}\)
then, trace of A or tr(A) = 2 + 4 = 6
Properties of Trace of a Matrix
Let A = \([a_{ij}]_{n\times n}\) and B = \([b_{ij}]_{n\times n}\) and \(\lambda\) be a scalar then
(i) tr(\(\lambda A\)) = \(\lambda\) tr(A)
(ii) tr(A + B) = tr(A) + tr(B)
(iii) tr(AB) = tr(BA)
Orthogonal Matrix
A square matrix is said to be orthogonal matrix if
\(AA^T\) = I (Identity matrix)
Note : The determinant value of orthgonal matrix is 1 or -1.
Example : Show that the matrix A = \(\begin{bmatrix} cosx & sinx \\ -sinx & cosx \end{bmatrix}\) is a orthogonal matrix.
Solution : We have,
A = \(\begin{bmatrix} cosx & sinx \\ -sinx & cosx \end{bmatrix}\)
\(A^{T}\) = \(\begin{bmatrix} cosx & -sinx \\ sinx & cosx \end{bmatrix}\)
Now, we have to find \(AA^T\) = \(\begin{bmatrix} cosx & sinx \\ -sinx & cosx \end{bmatrix}\)\(\begin{bmatrix} cosx & -sinx \\ sinx & cosx \end{bmatrix}\)
= \(\begin{bmatrix} cos^2x + sin^2x & -cosx.sinx + sinx.cosx \\ -sinx.cosx + sinx.cosx & cos^2x + sin^2x \end{bmatrix}\)
= \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\) = I (Identity matrix)
\(\implies\) \(AA^T\) = I
Hence, it is an orthogonal matrix.