The Identity Matrix and Diagonal Matrices

Another way to state this definition is that a square matrix is said to diagonalizable if and only if there exists some invertible matrix and diagonal matrix such that .

Probably the simplest way to see this is true is to take the determinant of the diagonal matrix. We can take the determinant of a diagonal matrix by simply multiplying all of the entries along its main diagonal. Since one of these entries is , then the determinant is , and hence the matrix is not invertible.

is the trace operation. It means to add up the entries along the main diagonal of the matrix. Since has ones along its main diagonal, the trace of is .

You can verify this directly by proving it, or by multiplying a few examples on your calculator.

There is no need to perform any elementary row operations on the identity matrix; it is already in its reduced row echelon form. (There is a leading one in each row, and each column).

A matrix is diagonal if and only if - that is, the element in column , row is equal to zero - for all . The given matrix violates this condition, since and five other elements are equal to nonzero numbers.

This is the correct term. Elementary matrices themselves can be used in place of elementary row operations when row reducing other matrices when convenient.

A matrix is diagonal if and only if - that is, the element in column , row is equal to zero - for all . The given matrix satisfies this condition, since its only nonzero elements are the first element in Column 1, the second element in Column 2, and so forth.

A matrix is diagonal if and only if - that is, the element in column , row is equal to zero - for all . The given matrix fits this criterion.

We can find the eigenvalues of the identity matrix by finding all values of such that .

Hence we have

So is the only eigenvalue, regardless of the size of the identity matrix.