Chapter 8: Problem 3
If \(P\) is orthogonal, show that \(k P\) is orthogonal if and only if \(k=1\) or \(k=-1\).
Short Answer
Expert verified
\(kP\) is orthogonal if \(k = 1\) or \(k = -1\).
Step by step solution
01
Definition of Orthogonal Matrix
A matrix \(P\) is orthogonal if its transpose is equal to its inverse. Mathematically, this is expressed as \(P^T = P^{-1}\). This implies that \(P^T P = I\), where \(I\) is the identity matrix.
02
Consider kP for Orthogonality
To determine if \(kP\) is orthogonal, compute \((kP)^T(kP)\) and check if it equals the identity matrix \(I\).
03
Compute Transpose of kP
The transpose of \(kP\) is \((kP)^T = kP^T\) because transposing involves linear scaling.
04
Compute (kP)^T(kP)
Substituting from the previous step, \((kP)^T(kP) = (kP^T)(kP) = k^2(P^T P)\). Since \(P^T P = I\), we have \(k^2 I = I\).
05
Solve k²I = I
For \(k^2 I = I\) to hold true, each element must satisfy \(k^2 = 1\). Thus, \(k = 1\) or \(k = -1\).
06
Conclusion about kP
Therefore, \(kP\) is orthogonal if and only if \(k = 1\) or \(k = -1\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Matrix Transpose
A matrix transpose involves flipping a matrix over its diagonal, effectively switching its rows with columns. For a given matrix \( A \), its transpose is denoted as \( A^T \). Here’s a simple breakdown:
When dealing with scalar multiplication, such as a matrix multiplied by a constant \( k \), the transpose of \( kA \) is \( kA^T \). This property is consistent due to the distributive nature of multiplication over addition in matrices. Knowing this is critical because when assessing the orthogonality of matrix forms like \( kP \), the transpose operation remains crucial.
Understanding the transpose operation is helpful for recognizing patterns in matrices and simplifying expressions, particularly in equations involving orthogonal matrices which have the property that \( P^T = P^{-1} \).
- Each element \( a_{ij} \) in matrix \( A \) becomes \( a_{ji} \) in \( A^T \).
- The first row becomes the first column, the second row becomes the second column, and so on.
When dealing with scalar multiplication, such as a matrix multiplied by a constant \( k \), the transpose of \( kA \) is \( kA^T \). This property is consistent due to the distributive nature of multiplication over addition in matrices. Knowing this is critical because when assessing the orthogonality of matrix forms like \( kP \), the transpose operation remains crucial.
Understanding the transpose operation is helpful for recognizing patterns in matrices and simplifying expressions, particularly in equations involving orthogonal matrices which have the property that \( P^T = P^{-1} \).
Matrix Inverse
The inverse of a matrix \( A \), denoted as \( A^{-1} \), is a matrix that when multiplied by \( A \), yields the identity matrix \( I \). Specifically, \( AA^{-1} = I \) and \( A^{-1}A = I \). Not all matrices have inverses, but if one exists, it’s unique.
Understanding how to find and verify inverses is crucial, especially in relation to orthogonal matrices, where we find this intriguing property: the inverse of an orthogonal matrix is also its transpose, i.e., \( P^{-1} = P^T \).
This interconnectedness significantly simplifies problems involving orthogonal matrices. Having \( P^T = P^{-1} \) shows us that orthogonal matrices save us from the exhaustive process of computing traditional and complex matrix inverses during many operations.
Understanding how to find and verify inverses is crucial, especially in relation to orthogonal matrices, where we find this intriguing property: the inverse of an orthogonal matrix is also its transpose, i.e., \( P^{-1} = P^T \).
This interconnectedness significantly simplifies problems involving orthogonal matrices. Having \( P^T = P^{-1} \) shows us that orthogonal matrices save us from the exhaustive process of computing traditional and complex matrix inverses during many operations.
Identity Matrix
The identity matrix \( I \) is the matrix equivalent of "1" in multiplication. It is a square matrix with ones on the diagonal and zeros elsewhere.
For matrices, multiplying by the identity matrix returns the original matrix, much like any number multiplied by one is unchanged. Mathematically, if \( A \) is a suitable sized matrix, \( AI = IA = A \).
In the context of orthogonal matrices such as \( P \), the relationship \( P^T P = I \) is vital. This confirms that the transpose multiplied by the matrix itself results in the identity matrix, maintaining the essence of orthogonality.
Identity matrices serve as crucial tools in matrix proofs and simplifications, especially for highlighting key properties like those seen with orthogonal matrices, where they assure the preservation of dimensions and correctness in matrix algebra.
For matrices, multiplying by the identity matrix returns the original matrix, much like any number multiplied by one is unchanged. Mathematically, if \( A \) is a suitable sized matrix, \( AI = IA = A \).
In the context of orthogonal matrices such as \( P \), the relationship \( P^T P = I \) is vital. This confirms that the transpose multiplied by the matrix itself results in the identity matrix, maintaining the essence of orthogonality.
Identity matrices serve as crucial tools in matrix proofs and simplifications, especially for highlighting key properties like those seen with orthogonal matrices, where they assure the preservation of dimensions and correctness in matrix algebra.
Scalar Multiplication
Scalar multiplication involves multiplying each element of a matrix by a constant. If \( A \) is a matrix and \( k \) is a scalar, then \( kA \) means each component of \( A \) is scaled by \( k \).
This straightforward operation becomes particularly important in linear algebra problems such as scaling matrices while maintaining their properties, or determining conditions for orthogonality.
With orthogonal matrices, the matrix \( kP \) remains orthogonal only if the scalar \( k \) satisfies specific criteria, in this case, \( k = 1 \) or \( k = -1 \). This preserves its inherent orthogonal properties without altering the matrix's fundamental relationship with the identity matrix via transposition.
Grasping scalar multiplication is fundamental to envisioning how matrices expand or contract with scalar influence, without affecting key properties like orthogonality when the criteria are well understood.
This straightforward operation becomes particularly important in linear algebra problems such as scaling matrices while maintaining their properties, or determining conditions for orthogonality.
With orthogonal matrices, the matrix \( kP \) remains orthogonal only if the scalar \( k \) satisfies specific criteria, in this case, \( k = 1 \) or \( k = -1 \). This preserves its inherent orthogonal properties without altering the matrix's fundamental relationship with the identity matrix via transposition.
Grasping scalar multiplication is fundamental to envisioning how matrices expand or contract with scalar influence, without affecting key properties like orthogonality when the criteria are well understood.
