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In a tube, the electron-emitting electrode is the cathode The cathode is usually heated by means of a wire filament, similar to the glowing element in an incandescent bulb The heat drives electrons from the cathode The cathode of a tube is analogous to the source of an FET, or to the emitter of a bipolar transistor The electron-collecting electrode is the anode, also called the plate The plate is the tube counterpart of the drain of an FET or the collector of a bipolar transistor In most tubes, intervening grids control the flow of electrons from the cathode to the plate The grids are the counterparts of the gate of an FET or the base of a bipolar transistor

1 Compute the cofactor matrix 2 Compute the inverse matrix Exercise B10 Use the properties of determinants to show that A 1 = 1 |A| 0 c Find (sI A) 1 s+f

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Directly Heated Cathode In some tubes, the filament also serves as the cathode This type of electrode is called a directly heated cathode The negative supply voltage is applied directly to the filament The filament voltage for most tubes is 6 V or 12 V dc It is important that dc be used to heat the filament in this type of tube, because ac will tend to modulate the output The schematic symbol for a diode tube with a directly heated cathode is shown in Fig 29-2A Indirectly Heated Cathode In many types of tubes, the filament is enclosed within a cylindrical cathode, and the cathode gets hot from the IR radiated by the filament This is an indirectly heated cathode The filament normally receives 6 V or 12 V ac or dc In an indirectly heated cathode arrangement, ac does not cause modulation problems, as it can with a directly heated cathode tube The schematic symbol for a diode tube with an indirectly heated cathode is shown in Fig 29-2B Because the electron emission in a tube depends on the filament or heater, tubes need a certain amount of time to warm up before they can operate properly This time can vary from a few seconds

Exercise B12 Verify Lemma B51 by direct multiplication of the righthand sides of eqns (B30) and (B31) Exercise B13 1 Let i and vi for i [1, n] denote the eigenvalues and eigenvectors of A Rn n Determine the eigenvalues and eigenvectors of A2 2 Determine the eigenvalues and eigenvectors of Am for any positive integer m 3 Use the above property of eigenvalues to show that the only possible eigenvalues of an idempotent matrix are 0 and 1 Exercise B14 For the special case of x Rn , either use direct multiplication or the properties of scalar and vector products to show that x Ba x = 0, where Ba is an antisymmetric matrix Exercise B15 Prove the following: 1 If B is a positive de nite matrix, then all the eigenvalues of B are strictly positive real numbers 2 If B is a positive semi-de nite matrix, then all the eigenvalues of B are non-negative real numbers 3 If B is a positive de nite matrix, then B 1 exists and is also positive de nite Exercise B16 Prove eqn (B33) by following the outline below Note that the hats on the variables have been dropped to simplify the notation 1 Consider the cost function J(Ao , ) = tr (Ao A) (Ao A) + Ao Ao I where the Lagrange multiplier is a symmetric matrix Take the derivative of J with respect to Ao and show that the matrix Ao that minimizes this expression is Ao = A (I + )

(B62)

symbol for diode tube with directly heated cathode At B, symbol for diode tube with indirectly heated cathode At C, simplified rendition of the construction of a diode tube

(B63)

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