Question

Let \(A\) and \(B\) be \(n \times n\) diagonal matrices (all entries off the main diagonal are zero). a. Show that \(A B\) is diagonal and \(A B=B A\). b. Formulate a rule for calculating \(X A\) if \(X\) is \(m \times n\). c. Formulate a rule for calculating \(A Y\) if \(Y\) is \(n \times k\).

Short answer

a. \(AB\) is diagonal and \(AB = BA\). b. Columns of \(X\) are scaled by \(A\)'s diagonal. c. Rows of \(Y\) are scaled by \(A\)'s diagonal.

Step-by-step solution

Understand Diagonal Matrices

A diagonal matrix is one where all the non-zero elements are located on the main diagonal from the top left to the bottom right. This means if we have an \(n \times n\) matrix, then any element \(a_{ij}\) where \(i eq j\) is zero.

Show AB is Diagonal

To multiply two diagonal matrices \(A\) and \(B\), consider that only the diagonal elements \(a_{ii}\) and \(b_{ii}\) will contribute to the product. Therefore, \(c_{ij} = a_{ii} \cdot b_{jj}\) when \(i = j\), and \(c_{ij} = 0\) for \(i eq j\). Thus, \(AB\) is also diagonal.

Show AB = BA

For diagonal matrices \(A\) and \(B\), the multiplication \((AB)_{ij}\) gives \(a_{ii}b_{ii}\), likewise \((BA)_{ij}\) also gives \(b_{ii}a_{ii}\). Therefore, \(AB = BA\), both being diagonal matrices with the same diagonal elements.

Calculate XA with X as m x n

For an \(m \times n\) matrix \(X\) and an \(n \times n\) diagonal matrix \(A\), the product \(XA\) results in scaling the columns of \(X\) by the diagonal elements of \(A\). Specifically, the \(j\)-th column of \(XA\) is given by multiplying the \(j\)-th column of \(X\) by \(a_{jj}\).

Calculate AY with Y as n x k

For an \(n \times k\) matrix \(Y\) and an \(n \times n\) diagonal matrix \(A\), the product \(AY\) results in scaling the rows of \(Y\) by the corresponding diagonal elements of \(A\). Specifically, the \(i\)-th row of \(AY\) is given by multiplying the \(i\)-th row of \(Y\) by \(a_{ii}\).

Key concepts

Matrix Multiplication

Matrix multiplication might seem daunting at first, but it's quite systematic once we break it down. In this operation, each element of the resulting matrix is calculated by taking the dot product of the corresponding row of the first matrix and the column of the second matrix. However, when working with diagonal matrices, the process simplifies significantly.

When multiplying two diagonal matrices, say \(A\) and \(B\), we notice that only the elements on the main diagonal need multiplication. This is because all the off-diagonal elements in a diagonal matrix are zeros. For matrices \(A\) and \(B\) where:

• \(a_{ij} = 0\) for \(i eq j\)
• \(b_{ij} = 0\) for \(i eq j\)

The product, \(C = AB\), remains diagonal, where each diagonal element \(c_{ii} = a_{ii}b_{ii}\). This property can drastically reduce computational complexity in calculations with diagonal matrices.

Commutative Property of Matrices

In general, matrix multiplication is not commutative meaning that for two matrices \(P\) and \(Q\), it is not always true that \(PQ = QP\). However, for diagonal matrices, this rule has a nice exception.

Given two \(n \times n\) diagonal matrices \(A\) and \(B\), we see that both \(AB\) and \(BA\) result in the same diagonal matrix. This happens because each individual multiplication involves simply the corresponding diagonal entries of the two matrices:

• \((AB)_{ii} = a_{ii}b_{ii}\)
• \((BA)_{ii} = b_{ii}a_{ii} = a_{ii}b_{ii}\)

Thus, \(AB = BA\), making the multiplication of diagonal matrices commutative. This can be especially useful in matrix algebra, making certain operations more flexible.

Scaling in Matrix Operations

Scaling in matrix operations refers to the multiplication of a matrix by a constant, or in the context of diagonal matrices, multiplying a matrix by another matrix that scales its rows or columns.

Consider scaling an \(m \times n\) matrix \(X\) by an \(n \times n\) diagonal matrix \(A\). The result, \(XA\), involves scaling each column of \(X\) by the corresponding diagonal element of \(A\). Specifically, the \(j\)-th column of \(XA\) is the \(j\)-th column of \(X\) scaled by \(a_{jj}\).

Similarly, if you have an \(n \times k\) matrix \(Y\) and multiply it by an \(n \times n\) diagonal matrix \(A\) to get \(AY\), this operation scales the rows of \(Y\). Each \(i\)-th row of \(Y\) is multiplied by the diagonal element \(a_{ii}\).

These scaling operations help efficiently modify matrices when working with linear transformations, proving to be both computationally simple and powerful in applications.