Real matrices A and B are said to be orthogonally equivalent if there exists an orthogonal matri

Real matrices A and B are said to be orthogonally equivalent...

Key Concepts

Orthogonal Matrix

An orthogonal matrix is a square matrix whose columns (and rows) are orthonormal vectors. This means that when the matrix is multiplied by its transpose, it results in the identity matrix. Orthogonal matrices preserve lengths and angles, making them essential in various transformations, rotations, and reflections. They play a crucial role in many areas of linear algebra, such as stability analysis, numerical methods, and matrix decompositions.

Equivalence Relation

An equivalence relation on a set is a binary relation that is reflexive, symmetric, and transitive. Reflexivity means every element is related to itself. Symmetry implies that if one element is related to another, then the second is related to the first. Transitivity ensures that if one element is related to a second, and the second is related to a third, then the first is related to the third. These properties partition the set into equivalence classes and are fundamental in understanding structural similarities across objects in mathematics.

Orthogonal Equivalence

Orthogonal equivalence between matrices is a specific type of equivalence relation where two matrices are considered equivalent if one can be transformed into the other via an orthogonal similarity transformation. In other words, two matrices A and B are orthogonally equivalent if there exists an orthogonal matrix P such that B = P?AP. This concept is particularly useful in simplifying problems in linear algebra by reducing matrices to simpler, canonical forms while preserving important properties like eigenvalues.

Transcript

00:02 We're told that two real matrices a and b are said to be orthogynally equivalent if there exists an orthogonal matrix p such that b is equal to p transpose ap.

00:14 We're asked to show that this relation is an equivalence relation on real matrices.

00:21 To prove it's an equivalence relation, we'll want to prove that it is a reflexive relation, a symmetric relation, and a transitive relation.

00:36 Prove it is reflexive.

00:39 Well, notice that if a is a real matrix, then it follows that a is equal to a, which is the same as i transpose a, moreover, i is in fact an orthogonal matrix.

01:15 Its rows are mutually orthogonal and are unit vectors.

01:26 Therefore, it follows that matrices a and a are orthogonally equivalent.

01:43 Therefore, it follows that the relation is reflexive.

01:51 Now we want to show the relation is symmetric.

01:54 Well, we'll suppose that a and b are real matrices, and we'll also suppose that a and b are orthogonally equivalent.

02:18 That means that there exists an orthogonal matrix p, such that b is equal to p transpose times a times p now this implies this implies that a is equal to well we know that because p is orthogonal p transpose equals p inverse solving for a we get a is equal to the inverse of p transpose times b times the inverse of p then using interchangeability of transpose and inverse, this is the inverse of the transpose of p times b times p transpose.

03:24 P transpose.

03:26 You know what? i'm sorry.

03:35 I'm not going to do that.

03:36 This is p inverse...

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