The trace of a 3 by 3 matrix A is defined as the summation of n=3;i=1;aii.
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The trace of an nxn matrix is usually thought of as the sum of the diagonal entries in the matrix. However, it is also the sum of the eigenvalues. This may help to understand why the proof works. So to answer your question, let's say A and B are matrices and A is similar to B. You want to prove that Trace A=Trace B If A is similar to B, there exists an invertible matrix P such that A=(P^-1 B P) Now we use the fact that Trace (AB)= Trace(BA) for any nxn matrices A and B.This is easy to prove directly from the definition of trace. (ask me if you need to know) So using this we have the following: Trace(A)=Trace(P^-1 B P)=Trace (BPP^-1)=Trace(B) and we are done! Dr. Chuck
The null matrix is also called the zero matrix. It is a matrix with 0 in all its entries.
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The first matrix has 3 rows and 2 columns, the second matrix has 2 rows and 3 columns. Two matrices can only be multiplied together if the number of columns in the first matrix is equal to the number of rows in the second matrix. In the example shown there are 3 rows in the first matrix and 3 columns in the second matrix. And also 2 columns in the first and 2 rows in the second. Multiplication of the two matrices is therefore possible.
The Transpose of a MatrixThe matrix of order n x m obtained by interchanging the rows and columns of the m X n matrix, A, is called the transpose of A and is denoted by A' or AT.