Are the elements of an averaging kernel matrix always centered on the diagonal, or can they be exclusively off-diagonal in rare cases?

Asked by: Erin Colon

What is an averaging kernel?

The averaging kernel matrix describes both the vertical resolution of each level in the retrieval and the relative weighting between the true profile and a priori profile.

Can a diagonal matrix have a zero on the diagonal?

A diagonal matrix is defined as a square matrix in which all off-diagonal entries are zero. (Note that a diagonal matrix is necessarily symmetric.) Entries on the main diagonal may or may not be zero.

When all diagonal elements of a matrix are equal?

Unit matrix is a diagonal matrix in which all the diagonal elements are unity.

Is identity matrix a diagonal matrix?

An identity matrix is a diagonal matrix in which all of the diagonal elements are equal to . No worries!

What are the conditions for diagonal matrix?

A square matrix in which every element except the principal diagonal elements is zero is called a Diagonal Matrix. A square matrix D = [d_ij]_{n x n} will be called a diagonal matrix if d_ij = 0, whenever i is not equal to j.

What is the off diagonal of a matrix?

In a table of numbers that has the same number of rows as columns, the entries that are not in the Main Diagonal are referred to as the off-diagonal entries in the table. In this example, all the 0s are in the off-diagonal cells.

What is the determinant of a matrix whose diagonal elements are zero?

By rule 6 the zero row means a zero determinant. This means: When a triangular matrix is singular (because of a zero on the main diagonal) its determinant is zero. All singular matrices have a zero determinant.

Is unit matrix is always diagonal matrix?

It is given that a unit matrix is a diagonal matrix in which all the diagonal elements are unity and all the other elements are zero. And all the other elements are equal to zero. Hence, the given statement, i.e. Unit matrix is a diagonal matrix in which all the diagonal elements are unity.

What is difference between diagonal and principal diagonal?

A diagonal matrix is a square matrix in which all the elements that are NOT in the principal diagonal are zeros and the elements of the principal diagonal can be either zeros or non-zeros.

Is diagonal matrix always a square matrix?

And for this question, the important thing to realize is that all diagonal matrices are square matrices. Therefore, just from this definition, we can see that our statement is false. A diagonal matrix does in fact have to be a square matrix.

Is a diagonal matrix always invertible?

Proposition A diagonal matrix is invertible if and only if all the entries on its main diagonal are non-zero. A diagonal matrix is triangular and a triangular matrix is invertible if and only if all the entries on its main diagonal are non-zero.

What is the difference between scalar and diagonal matrix?

The only difference between the scalar matrix and a diagonal matrix is the elements of the principal diagonal. In a scalar matrix, the elements of the principal diagonal are all equal to the same constant value, and in a diagonal matrix the principal diagonal elements are all of different values.

Are all diagonal matrices orthogonal?

Every diagonal matrix is orthogonal.

Is a matrix with zeros on the diagonal invertible?

Matrix with zeros on diagonal and ones in other places is invertible.

Is matrix with 0 diagonal invertible?

If any element of the main diagonal of a diagonal matrix is zero, then the entire row of the matrix becomes zero, which makes the matrix singular, and hence, the inverse of the diagonal matrix won’t exist.

Can we say null matrix a diagonal matrix?

Statement 1: Every null matrix of order n is a diagonal matrix. ∵ All the principal diagonal elements of matrix O are zero which violates the definition of a diagonal matrix. As for a diagonal matrix all the elements are zero except those in the principal diagonal. Hence, statement 1 is false.

Can any matrix be diagonalized?

Hence, a matrix is diagonalizable if and only if its nilpotent part is zero. Put in another way, a matrix is diagonalizable if each block in its Jordan form has no nilpotent part; i.e., each “block” is a one-by-one matrix.

What makes a matrix not diagonalizable?

If there are fewer than n total vectors in all of the eigenspace bases B λ , then the matrix is not diagonalizable.

Is every diagonal matrix diagonalizable?

Matrix diagonalization is the process of performing a similarity transformation on a matrix in order to recover a similar matrix that is diagonal (i.e., all its non-diagonal entries are zero). Once a matrix is diagonalized it becomes very easy to raise it to integer powers. Not all matrices are diagonalizable.