A square matrix A is called orthogonal if_______ where A’ is the transpose of A.
A’ = A -1
Understanding Orthogonal Matrices
A square matrix is a matrix that has the same number of rows and columns. Matrices have special properties and types, one of which is an orthogonal matrix. The question asks for the condition that defines an orthogonal matrix.
Definition of an Orthogonal Matrix
A square matrix A is called an orthogonal matrix if its transpose is equal to its inverse. The transpose of a matrix A, denoted by A’ or $A^T$, is obtained by switching the rows and columns of A. The inverse of a matrix A, denoted by $A^{-1}$, is a matrix such that when multiplied by A, it results in the identity matrix (if the inverse exists).
The defining condition for a square matrix A to be orthogonal is:
$$ A’ = A^{-1} $$
This condition is also equivalent to:
$$ AA’ = A’A = I $$
where I is the identity matrix of the same size as A.
Analyzing the Given Options
Let's look at the provided options and compare them to the definition of an orthogonal matrix:
| Option | Condition | Is it the definition of an Orthogonal Matrix? |
|---|---|---|
| 1 | $A = A^2$ | No, this condition defines an idempotent matrix. |
| 2 | $A’ = A^{-1}$ | Yes, this is the standard definition of an orthogonal matrix. |
| 3 | $A = A^{-1}$ | No, this means A squared is the identity matrix ($A^2 = I$). Such a matrix is called an involutory matrix. |
| 4 | $A = A’$ | No, this condition defines a symmetric matrix. |
Based on the analysis, the condition $A’ = A^{-1}$ is the correct definition of an orthogonal matrix.
Summary of Orthogonal Matrix Properties
Here are some key properties of orthogonal matrices:
- They are always square matrices.
- Their columns (and rows) form an orthonormal set of vectors. This means the dot product of any two distinct columns is zero, and the dot product of a column with itself is one.
- The determinant of an orthogonal matrix is either +1 or -1.
- Orthogonal matrices preserve the Euclidean norm (length) of vectors and the dot product between vectors. They represent transformations like rotations, reflections, or a combination of both in geometric terms.
- The inverse of an orthogonal matrix is also orthogonal.
- The product of two orthogonal matrices is orthogonal.
Therefore, the correct condition for a square matrix A to be orthogonal is when its transpose A’ is equal to its inverse $A^{-1}$.
Revision Table: Matrix Types and Definitions
| Matrix Type | Defining Condition | Description |
|---|---|---|
| Orthogonal Matrix | $A’ = A^{-1}$ (or $AA’ = I$) | Transpose equals inverse; preserves lengths and angles. |
| Symmetric Matrix | $A = A’$ | Matrix is equal to its transpose. |
| Idempotent Matrix | $A = A^2$ | Multiplying the matrix by itself yields the original matrix. |
| Involutory Matrix | $A = A^{-1}$ (or $A^2 = I$) | Multiplying the matrix by itself yields the identity matrix. |
| Identity Matrix (I) | Square matrix with 1s on the main diagonal and 0s elsewhere. | Multiplying any matrix by the identity matrix (of compatible size) leaves the matrix unchanged. |
| Transpose ($A’$ or $A^T$) | Rows become columns and columns become rows. | Operation performed on a matrix. |
| Inverse ($A^{-1}$) | $AA^{-1} = A^{-1}A = I$ | Matrix that, when multiplied by the original matrix, yields the identity matrix. Not all matrices have an inverse. |
Additional Information: Significance of Orthogonal Matrices
Orthogonal matrices are very important in various fields, including linear algebra, geometry, physics, and computer graphics. Their property of preserving lengths and angles makes them suitable for describing rotations and reflections without distortion. For example, in 3D computer graphics, rotation matrices used to orient objects are typically orthogonal. In quantum mechanics, unitary matrices (which are complex analogues of orthogonal matrices) play a similar role.
Understanding the definition $A’ = A^{-1}$ is fundamental to recognizing and working with orthogonal matrices in mathematical and applied contexts. This condition implies that the columns (and rows) of an orthogonal matrix form a basis of orthonormal vectors, which is a set of vectors that are mutually perpendicular and each have a length of 1.
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