Question

A square matrix \(P\) is called an idempotent if \(P^{2}=P .\) Show that: a. 0 and \(I\) are idempotents. b. \(\left[\begin{array}{cc}1 & 1 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}1 & 0 \\ 1 & 0\end{array}\right],\) and \(\frac{1}{2}\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right],\) are idem- c. If \(P\) is an idempotent, so is \(I-P\). Show further that \(P(I-P)=0\). d. If \(P\) is an idempotent, so is \(P^{T}\). e. If \(P\) is an idempotent, so is \(Q=P+A P-P A P\) for any square matrix \(A\) (of the same size as \(P\) ). f. If \(A\) is \(n \times m\) and \(B\) is \(m \times n,\) and if \(A B=I_{n}\), then \(B A\) is an idempotent.

Short answer

0, I, and specified matrices are idempotents; I-P, P^T, Q, and BA are idempotents if conditions are met.

Step-by-step solution

Show 0 and Identity Matrix are Idempotents

A zero matrix, denoted as 0, is idempotent since its square is zero. Hence, \(0^2 = 0\). The identity matrix \(I\) is idempotent because its square is itself, \(I^2 = I\). These follow directly from the definitions of 0 and \(I\) as matrices.

Verify Specific Matrices are Idempotents

For \( P = \begin{bmatrix} 1 & 1 \ 0 & 0 \end{bmatrix} \), calculate \( P^2 = P \begin{bmatrix} 1 & 1 \ 0 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 1 \ 0 & 0 \end{bmatrix} \), showing idempotency. Likewise, for \( \begin{bmatrix} 1 & 0 \ 1 & 0 \end{bmatrix} \), compute the square matrix to confirm it's idempotent. Lastly, \( \frac{1}{2} \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \) squared gives the original matrix, confirming idempotency.

Prove I-P is Idempotent When P is Idempotent

If \(P\) is idempotent, then \(I - P\) also is. We show \((I - P)^2 = I - P\):\[(I-P)(I-P) = I - 2P + P^2 = I - 2P + P = I - P\]. Also, \(P(I - P) = P - P^2 = P - P = 0\).

Show the Transpose of an Idempotent Matrix is Also Idempotent

Given \(P^2 = P\), then transpose both sides: \((P^2)^T = P^T\). By matrix property, \((AB)^T = B^T A^T\), hence \(P^T P^T = P^T\). So \(P^T\) is idempotent.

Verify Q = P + AP - PAP is Idempotent When P is Idempotent

Prove \(Q^2 = Q\):Let \(Q = P + AP - PAP\), compute \(Q^2 = (P + AP - PAP)^2 = P + 2AP - PAP - PAP = P + AP - PAP = Q\).Hence, \(Q\) is idempotent.

Show BA is Idempotent When AB Equals Identity

Given \(AB = I\) where \(A \in \mathbb{R}^{n \times m}\) and \(B \in \mathbb{R}^{m \times n}\). We check \((BA)^2 = BA\): from \((BA)(BA) = B(AB)A = BI_nA = BA\), proving \(BA\) is idempotent.

Key concepts

Matrix algebra

Matrix algebra, a fundamental element of linear algebra, involves operations with matrices. These operations include addition, subtraction, and multiplication, among others. It is a crucial tool in various applications such as computer graphics, physics, and statistics. Matrices are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns.

In matrix algebra, matrices must adhere to specific rules, especially for multiplication, which isn't merely elementwise, but follows a row-by-column formula. Two matrices can only be multiplied if the number of columns in the first matrix matches the number of rows in the second. The resulting product has dimensions governed by the rows of the first and the columns of the second matrix. This adherence to rules forms the basis of understanding more complex matrix behaviors, such as idempotency.

Idempotent matrices in this context are matrices which, when squared, yield the same matrix again. This property is central to the original exercise and illustrates an interesting aspect of matrix algebra.

Matrix transpose

Matrix transpose is an operation that flips a matrix over its diagonal. In the transpose of a matrix, the rows become columns and the columns become rows. If you have a matrix \( A \), the transpose of \( A \) is denoted as \( A^T \).

Transposing matrices is crucial for certain mathematical proofs and transformations. For example, in the exercise provided, if a matrix \( P \) is idempotent, so is its transpose \( P^T \). This step confirms that if \( P^2 = P \), then \( (P^2)^T = (P^T)^2 = P^T \). This result relies on the property that the transpose of a product of matrices is the product of their transposes in reverse order, i.e., \( (AB)^T = B^T A^T \).

Understanding the transpose is vital for many linear algebra processes, such as solving systems of equations or facilitating matrix decompositions.

Matrix multiplication

Matrix multiplication is a cornerstone concept in matrix algebra that extends beyond simple arithmetic. It involves taking the row elements of one matrix and multiplying them with the column elements of another. The results are then summed to create a single element in the resulting matrix.

In the context of idempotent matrices, the matrix \( P \) multiplies with itself to verify its idempotency, i.e., \( P^2 = P \). Consistency in matrix multiplication rules confirms properties like these, since each element of the resulting matrix is a sum of products, calculated across corresponding rows and columns from the original matrices.

Remember, matrix multiplication is not commutative; \( AB \) does not necessarily equal \( BA \). This non-commutative nature emphasizes the importance of the order in which matrices are multiplied, particularly in exercises dealing with multiple matrices like \( Q = P + AP - PAP \). Proper execution of these operations ensures accuracy in proving certain matrix properties.

Identity matrix

The identity matrix, denoted as \( I \), is a special type of square matrix. It acts as the multiplicative identity in matrix algebra, much like the number 1 does in arithmetic. Each element in the main diagonal is 1, and all other elements are 0.

In any matrix system, multiplying by an identity matrix leaves the original matrix unchanged, meaning \( AI = A \) and \( IA = A \) for any compatible matrix \( A \). This property is key in proving idempotency of matrices.

For example, if \( A \) and \( B \) are matrices such that \( AB = I_n \), then \( BA \) is an idempotent matrix, meaning \( (BA)^2 = BA \). This specific exercise illustrates how the identity matrix plays a significant role in simplifying and proving complex algebraic properties of matrices.