Provide a character analysis of Katharina in Shakespeare's The Taming of the Shrew.

In Shakespeare's The Taming of the Shrew, Katharina is an ill-tempered, ill-mannered, nasty, and rude young woman of exceptional beauty, who refuses to be courted by any of the young men of the town.

In Act Two, scene one, Katharina has tied her sister up and demands to know who she likes best of all her suitors. When Bianca's answers do not please her, she slaps her sister in the face.

KAT:

Of all thy suitors, here I charge thee, tell

Whom thou lovest best: see thou dissemble not. (8-9)

Bianca swears that no one man appeals to her any more than another.

BIA:

Believe me, sister, of all the men alive

I never yet beheld that special face

Which I could fancy more than any other. (10-12)

Bianca denies than any man appeals to her. She offers her anything she owns (jewelry, clothes, etc.), and even is willing to speak on Katharina's behalf if she desires a certain young man. Katharina believes it is all a bunch of lies, and smacks Bianca just as her father is entering the room.

KAT:

If that be jest, then all the rest was so.

Strikes her.

Baptista is the father of the two girls; he enters, comforting Bianca and chiding Katharina, but his oldest daughter will not be corrected by her father and promises to have her revenge against Bianca.

KAT:

Her silence flouts me, and I'll be revenged.

Katharina is also abusive toward the young men of the town who come to court her. However, Petruchio arrives from Verona and hears about Katharina. It seems that Baptista will not allow his younger and gentler daughter Bianca to marry until the nasty Katharina is wed. And Katharina is having nothing to do with this plan. Petruchio has come, he says, to find a wife. When he learns of Katharina whose father is offering a large dowry in marriage, but has a shrewish disposition, Petruchio ignores the warnings of those around him and promises that he will marry her for her money, unafraid of her ill-temper.

When Petruchio has made arrangements with Baptista to marry his oldest daughter, he asks Baptista to send Katharina to him. He addresses her, however, as "Kate." He speaks sweetly to her, trying to woo her, saying that she "moves" him. But as is her custom, Kate tells him to leave alluding that his is "movable"—like a piece of furniture, and nothing more:

KAT:

Moved! in good time: let him that moved you hither

Remove you hence: I knew you at the first

You were a moveable.

Her insults continue—she infers that he is an "ass," among other things; but Petruchio is undaunted, eventually forcing her to marry him despite the hatred she exhibits toward him.

At the start of the play, Katharina seems a spoiled brat, but reading between the lines, I believe she is miserable because all the young men prefer her younger sister and her father (Baptista) favors Bianca as well.

KAT:

… Nay, now I see

She is your treasure, she must have a husband;

I must dance bare-foot on her wedding day

And for your love to her lead apes in hell.

Talk not to me: I will go sit and weep

Till I can find occasion of revenge. (31-36)

Kate does not help her case by being so mean-spirited, but in a way, she may do so because acting nasty helps her to create a reason why the young men do not like her. She would rather believe it is her behavior that puts these men off, rather than believe it is because she is less worthy than her sister.

How is Katherina portrayed in The Taming of the Shrew?

Katherina is obviously the "shrew" referenced in William Shakespeare's The Taming of the Shrew, and she lives up to that reputation, at least in most ways. A shrew is defined as a "an unpleasant, bad-tempered woman" or a "an ill-tempered scolding woman." Either one is bad, and Katherina seems to be both.

The first time we meet her she is described this way:

 "That wench is stark mad or wonderful froward."

Katharina throws things, hits people, and screams a lot, and most of the people in her life are at least moderately afraid of her--most are flat-out terrified of her temper and her sharp tongue. She is the talk of the town because everyone knows what kind of person she is, and they also know that her father is afraid he will never be able to marry her off. That is evident by the fact that he has to resort to bribery, hoping someone who wants ot marry his younger daughter, Bianca, will be motivated to find someone willing to marry the shrewish Katherina. 

That is exactly what happens, of course, and Petruchio is unmoved by her tantrums, determined to "kill her with kindness." Every moment of their relationship seems designed to humiliate, frustrate, and hurt Katherina, and the method is effective until Katherina is eventually "tamed."

Those are the evident and obvious characteristics of Katherina; however, she is a much more complex character than just a raving harpie who is angry at everyone. Based on her sharp-tounged wit, it is obvious that she is a very bright young woman. She is also someone who demands respect, as demonstrated here:

Why, sir, I trust I may have leave to speak;
And speak I will; I am no child, no babe:
Your betters have endured me say my mind,
And if you cannot, best you stop your ears.
My tongue will tell the anger of my heart,
Or else my heart concealing it will break,
And rather than it shall, I will be free
Even to the uttermost, as I please, in words.

It is easy, when reading the play, to assume that Katherina has always just been this way, but it is likely that she has not. Her father has put her on the marriage auction block, so to speak, when he says to two of Bianca's suitors (neither of whom has any interest whatsoever in Katherina, as he well knows):

If either of you both love Katharina,
Because I know you well and love you well,
Leave shall you have to court her at your pleasure.

While it may appear that Bianca is "the good sister," I will remind you that she lies to and disobeys her father at every opportunity when there is something she wants (Hortensio); since she does it so willingly and easily, we must assume that this is a pattern with her. It is easy to see that Bianca has rather manipulated her father into seeing her in a positive light, something Katherina was not willing to do. She is human and she wanted her father's attention as much as Bianca did, but she was not willing to stoop to trickery and deceit to get it like Bianca was. Oddly enough, then, Katherina is the better person, and in the end, of course, Bianca is a much more disobedient wife than Katherina. 

Petruchio wins his battle to tame Katherina, but only because she lets him do so. This may be Katherina's story, but she is not the only shrew in the tale. The widow and Bianca both exhibit some pretty shrewish behavior at the banquet--they are harsh, scolding, and ill-tempered. Katherina's descent into "shrew-dom" was a reaction to her circumstances, and her ascent out of "shrew-dom" was a decision she made.