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Prove that trace of the matrix is invariant under similarity transformation?
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
What is the definition of a null matrix?
The null matrix is also called the zero matrix. It is a matrix with 0 in all its entries.
What is the inverse of the matrix 3?
It is the matrix 1/3It is the matrix 1/3It is the matrix 1/3It is the matrix 1/3
Is it possible to multiply a 3 X 2 matrix and a 2 X 3 matrix?
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.
What is the definition of transpose in regards to a matrix?
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.
Prove that trace of the matrix is invariant under similarity transformation?
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
What is the trace of a matrix to a power?
It is the diagonal entries of the matrix raised to a power.
What is the definition of involtary matrix?
Involtary Matrix A square matrix A such that A2=I or (A+I)(A-I)=0, A is called involtary matrix.
What is the definition of matrix?
The rectangular array of elements enclosed by pair of brackets, such as [ 1 2 3 ] and subject to certain rules of operation is called matrix.
What is the definition of a null matrix?
The null matrix is also called the zero matrix. It is a matrix with 0 in all its entries.
What is payroll matrix?
can anyone give me an exact definition of payroll matrix................
What is the definition of zero matrix?
Zero Matrix When all elements of a matrix are zero than the matrix is called zero matrix. Example: A=|0 0 0|
What is the definition of identity matrix?
Identity or Unit Matrix If in the scaler matrix the value of k=1, the matrix is called the identity or unit matrix. It is denoted by I or U.
Definition of Lower-triangular matrix?
Lower-triangular Matrix A square matrix A whose elements aij=0 for i
What is the definition of a diagonal matrix?
Diagonal Matrix A square matrix A which is both uper-triangular and lower triangular is called a diagonal matrix. Diagonal matrix is denoted by D.
What is the inverse of the matrix 3?
It is the matrix 1/3It is the matrix 1/3It is the matrix 1/3It is the matrix 1/3
What is the definition of an idempotent matrix?
A square matrix A is idempotent if A^2 = A. It's really simple
