Question

An \(n \times n\) matrix \(P\) is called an idempotent if \(P^{2}=P\). Show that: a. \(I\) is the only invertible idempotent. b. \(P\) is an idempotent if and only if \(I-2 P\) is selfinverse.

Short answer

a. Only \(I\) is an invertible idempotent matrix. b. \(P\) is idempotent iff \(I-2P\) is selfinverse.

Step-by-step solution

Understand the idempotent matrix definition

A matrix \(P\) is called idempotent if multiplying it by itself yields the same matrix, i.e., \(P^2 = P\). This means every entry in the resulting squared matrix is the same as in the original matrix.

Explore invertibility of idempotent matrices

For a matrix to be invertible, there must exist another matrix \(Q\) such that \(PQ = QP = I\), where \(I\) is the identity matrix. If \(P^2 = P\) and it is invertible, then \(P^{-1}P^2 = P^{-1}P\), yielding \(P = I\). Hence, \(I\) is the only invertible idempotent matrix because any deviation from \(I\) leads to a non-invertible matrix.

Show that if \(P\) is idempotent, \(I-2P\) is selfinverse

Assume \(P\) is idempotent, \(P^2 = P\). Check if \((I - 2P)^2 = I\):\[(I - 2P)^2 = (I - 2P)(I - 2P) = I - 2P - 2P + 4P^2\] Since \(P^2 = P\), substitute to get:\[ = I - 4P + 4P = I \].Thus, \(I-2P\) is indeed selfinverse.

Show that if \(I-2P\) is selfinverse, \(P\) is idempotent

Assume \((I-2P)^2 = I\) and check for idempotency:1. Expand \((I - 2P)^2\) to get the equation: \[I - 4P + 4P^2 = I\]. Simplifying this, \(4P^2 - 4P = 0\). 2. Factor out \(4P\) to obtain the expression \(P^2 - P = 0\). Therefore, \(P(P - I) = 0\) which implies \(P^2 = P\), confirming \(P\) is idempotent.

Key concepts

Matrix Invertibility

In the realm of matrices, a matrix is considered invertible if there exists another matrix, referred to as its inverse, which when multiplied together return the identity matrix.

More formally, for an invertible matrix \(P\), there must be a matrix \(Q\) such that \(PQ = QP = I\), where \(I\) is the identity matrix.

Invertible matrices are quite like the non-zero numbers in basic arithmetic that have reciprocals. If a matrix isn't invertible, it's often labeled as "singular," meaning there's no matrix that can "undo" its effects through multiplication.

When exploring idempotent matrices, if \(P^2 = P\), finding an inverse becomes intriguing. If such an idempotent is invertible, an interesting conclusion emerges: it must be identical to the identity matrix \(I\). This insight stems from the fact that any deviation away from \(I\) will result in a lack of an inverse. Only the identity matrix stands out as the sole invertible idempotent matrix.

Self-inverse Matrix

A matrix is classified as self-inverse, or involutory, if multiplying it by itself results in the identity matrix. Mathematically, for a self-inverse matrix \(A\), it holds that \(A^2 = I\).

This property implies that applying the matrix twice reverses any initial transformation, akin to taking two steps forward and two steps backward, ending up where you started.

In context with idempotent matrices, a self-inverse matrix can be crafted as \(I-2P\) where \(P\) is idempotent. This is established by exploring the expression \((I - 2P)^2 = I\), which simplifies to confirm the self-inverse nature.

Thus, for an idempotent matrix \(P\), \(I-2P\) naturally inherits the property of being self-inverse, showcasing a unique symmetry in matrix algebra.

Identity Matrix

The identity matrix operates as the backbone of matrix operations, acting like the number 1 in regular arithmetic. It is generally denoted by \(I\) and possesses a simplistic yet potent feature: multiplicative identity.

An \(n \times n\) identity matrix has ones on the diagonal and zeros elsewhere. When any matrix \(A\) is multiplied by the corresponding identity matrix, the result is unchanged, i.e., \(IA = AI = A\).

The identity matrix serves as a critical concept when discussing invertibility. In a landscape of idempotent matrices, \(I\) stands tall as the only one maintaining an inverse, fundamentally because it maintains its form when squared (\(I^2 = I\)).

Furthermore, in the problem exercise, the identity matrix also illustrates the linkage between idempotency and self-inverse matrices, where \(I - 2P\) sets up a compelling transformation towards an involutory form.

Matrix Algebra

Matrix algebra forms the foundation for dealing with matrices and their manipulation, providing rules and structures that facilitate many branches of mathematics and applied sciences.

It involves various operations, such as addition, subtraction, multiplication, and finding inverses, each adhering to specific rules, much like addition or multiplication in regular arithmetic.

A key operation in matrix algebra is multiplication. Unlike regular numbers, matrices must meet size compatibility to multiply. Associative and distributive properties are preserved, yet commutative properties do not necessarily hold.

Another essential aspect of matrix algebra is determinents which help ascertain matrix properties such as invertibility.

In dealing with idempotent and self-inverse matrices, matrix algebra facilitates identifying characteristics and establishes relationships through transformations. It provides a versatile framework to explore mathematical entities like idempotent transformations or formulating expressions such as self-inverses \((I - 2P)\) directly stemming from fundamental properties of matrices.