Geometric Bites

A diagonal matrix is a square matrix in which every entry away from the main (leading) diagonal is zero. The leading diagonal runs from the top-left corner of the matrix to the bottom-right corner, and these diagonal entries are the only positions that may contain non-zero values. All off-diagonal entries must be zero. The diagonal entries themselves can be any real numbers, including zero. This strict structure is what makes diagonal matrices especially simple to analyse and compute with in linear algebra. Examples of Diagonal Matrices The general 2×2 diagonal matrix has the form: (a 0) (0 b) The general 3×3 diagonal matrix has the form: (a 0 0) (0 b 0) (0 0 c) In both cases, the values on the leading diagonal (a, b, c, …) are the only entries that may be non-zero. Every position above or below this diagonal is fixed at 0. The General n×n Diagonal Matrix For an n×n dia...

Reversing a Linear Transformation Using an Inverse Matrix In linear algebra, any invertible linear transformation can be reversed. The key tool that makes this possible is the inverse matrix . If a matrix transforms a vector into another, the inverse matrix recovers the original. 1. The Transformation Equation Suppose a vector x₁ is transformed into a vector x₂ using a matrix T : T x₁ = x₂ This equation describes how x₁ is mapped to x₂ . To reverse the transformation, we must apply the inverse matrix. 2. Applying the Inverse Matrix Multiply both sides of the equation by T⁻¹ : T⁻¹ (T x₁) = T⁻¹ x₂ Using the fundamental identity: T⁻¹ T = I the expression simplifies directly to: x₁ = T⁻¹ x₂ 3. Interpretation This tells us that the original vector is obtained by applying the inverse matrix to the transformed vector: Original vector = Inverse matrix × Image vector As long as the matrix is invertible, the reverse transformation always exist...

What Is an Isomorphism? In mathematics, an isomorphism is a function that shows two mathematical objects have the same structure. Although the objects may look different, an isomorphism demonstrates that they behave in exactly the same way with respect to the operations that define them. If such a map exists, the objects are called isomorphic . An isomorphism tells us that two systems are essentially the same, differing only by a relabelling of their elements. The Basic Idea An isomorphism is a function: f : A → B that must be: Injective — different elements of A map to different elements of B. Surjective — every element of B comes from some element of A. Together these mean f is bijective , and so it has an inverse: f⁻¹ : B → A No information is lost moving from A to B or back. Preserving Structure Bijectivity alone is not enough. An isomorphism must also preserve structure. For groups, this means: f(a ⋆ b) = f(a) ∘ f(b) for all a, b ∈ A ,...