Q36E Consider a symmetric nxn聽matrix... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 8: Q36E (page 393)

Consider a symmetric nxnmatrix A with $A^{2} = A$ . Is the linear transformation $T (\overset{鈬赌}{x}) = A \overset{鈬赌}{x}$ necessarily the orthogonal projection onto a subspace of ${鈩�}^{n}$ ?

Short Answer

Expert verified

Yes, The linear transformation $T (\overset{鈬赌}{x}) = A \overset{鈬赌}{x}$ is an orthogonal projection onto the subspace of ${鈩�}^{n} ({ontoE}_{1})$ .

Step by step solution

Orthogonal projection:

The orthogonal projection of one vector onto another serves as the foundation for decomposing a vector into a sum of orthogonal vectors. A vector v's projection onto a second vector w is a scalar multiple of the vector w.

Find the linear transformation:

Here we have

$位 \overset{鈬赌}{v} = A \overset{鈬赌}{v} 鈫� 位 \overset{鈬赌}{v} = A^{2} \overset{鈬赌}{v} = 位^{2} \overset{鈬赌}{v} 位 = 位^{2} 位 - 位^{2} = 0 位 (1 - 位) = 0 Hence, 位 = 0 and 位 = 1$

Now, since the matrix A is symmetric, the eigenspaces $E_{0} and E_{1}$ corresponding to the eigenvalueslocalid="1659612821751" $位 = 0 and 位 = 1$ respectfully, are orthogonal complements. Now, $位 = 1$ is when the transformation T leaves the vector unchanged, and we have

$E_{1}^{鈯�} = E_{0}$ .

Also, vectors in $E_{1}^{鈯�}$ are mapped to $\overset{鈬赌}{0}$ by T.

Therefore, the linear transformation $T (\overset{鈬赌}{x}) = A \overset{鈬赌}{x}$ is an orthogonal projection onto the subspace of ${鈩�}^{n} (onto E_{1})$ .

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91影视

Consider a symmetric nxnmatrix A with $A^{2} = A$ . Is the linear transformation $T (\overset{鈬赌}{x}) = A \overset{鈬赌}{x}$ necessarily the orthogonal projection onto a subspace of ${鈩�}^{n}$ ?

Short Answer

Step by step solution

Orthogonal projection:

Find the linear transformation:

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91影视

Consider a symmetric nxnmatrix A withA2=A. Is the linear transformationT(x鈬赌)=Ax鈬赌necessarily the orthogonal projection onto a subspace of鈩�n?

Short Answer

Step by step solution

Orthogonal projection:

Find the linear transformation:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Geometry

Logic and Functions

Applied Mathematics

Calculus

Decision Maths

Study anywhere. Anytime. Across all devices.

Consider a symmetric nxnmatrix A with $A^{2} = A$ . Is the linear transformation $T (\overset{鈬赌}{x}) = A \overset{鈬赌}{x}$ necessarily the orthogonal projection onto a subspace of ${鈩�}^{n}$ ?