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Chapter 2: Q18Q (page 93)

Suppose P is invertible and \(A = PB{P^{ - 1}}\). Solve for Bin terms of A.

Short Answer

Expert verified

\(B = {P^{ - 1}}AP\)

Step by step solution

01

Condition for an invertible matrix

Theorem 5states that Ais an invertible \(n \times n\) matrix, then for each b in \({\mathbb{R}^n}\), the equation \(Ax = b\) has a unique solution \(x = {A^{ - 1}}b\).

02

Solve for B in terms of A

Multiply both sides of the equation \(A = PB{P^{ - 1}}\) by \({P^{ - 1}}\):

\(\begin{aligned}{c}{P^{ - 1}}A = {P^{ - 1}}PB{P^{ - 1}}\\{P^{ - 1}}A = IB{P^{ - 1}}\\{P^{ - 1}}A = B{P^{ - 1}}\end{aligned}\)

Multiply both sides of theobtainedequation by P:

\(\begin{aligned}{c}{P^{ - 1}}AP = B{P^{ - 1}}P\\{P^{ - 1}}AP = BI\\{P^{ - 1}}AP = B\end{aligned}\)

Thus, \(B = {P^{ - 1}}AP\).

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Most popular questions from this chapter

Suppose A is invertible. Explain why \({A^T}A\) is also invertible. Then show that \({A^{ - {\bf{1}}}} = {\left( {{A^T}A} \right)^{ - {\bf{1}}}}{A^T}\).

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21. The actual color a viewer sees on a screen is influenced by the specific type and amount of phosphors on the screen. So, each computer screen manufacturer must convert between the (R, G, B) data and an international CIE standard for color, which uses three primary colors, called X, Y, and Z. A typical conversion for short-persistence phosphors is

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A computer program will send a stream of color information to the screen, using standard CIE data (X, Y, Z). Find the equation that converts these data to the (R, G, B) data needed for the screen鈥檚 electron gun.

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\(A = \left( {\begin{aligned}{*{20}{c}}2&0&{ - 1}\\4&{ - 5}&2\end{aligned}} \right)\), \(B = \left( {\begin{aligned}{*{20}{c}}7&{ - 5}&1\\1&{ - 4}&{ - 3}\end{aligned}} \right)\), \(C = \left( {\begin{aligned}{*{20}{c}}1&2\\{ - 2}&1\end{aligned}} \right)\), \(D = \left( {\begin{aligned}{*{20}{c}}3&5\\{ - 1}&4\end{aligned}} \right)\) and \(E = \left( {\begin{aligned}{*{20}{c}}{ - 5}\\3\end{aligned}} \right)\)

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